
This here is the Byzantine Empire. No, not the green one. The purple one. Squint a bit and you'll see it. Our "empire" consists of two provinces, Thrace (the province with our capital, Constantinople), and Morea, in the south of Greece. Those little islands in the Aegean sea don't belong to us incidentally, despite being purple. That's Naxos, one of Venice's puppet states.
First things first, lets take a look at our budget.

The budget screen lets us see our income (poor) and our investments (underwhelming). Income in EU3 comes in two types: monthly income, and yearly income. Monthly income arrives at the beginning of each month, and gets put into the sliders on the right of the budget screen. Yearly income (or the census tax, as it's also called) arrives at the beginning of the year (January 1st) and goes straight to the treasury for us to spend on nice things.
Sticking with monthly income, we can distribute it to the five tech areas (Government, Production, Trade, Land, and Naval), Stability (a measure of our countries internal stability, measuring from -3 to +3), or the Treasury. Take special note of the treasury slider. It's to the far left. This is its natural habitat. Monthly income sent to the treasury can provide extra spending money in the short term, but it also causes gradual inflation, which drives up the cost of troops, buildings, tech, and... well pretty much everything. Building up some inflation is pretty much unavoidable, but you really don't want to get too much, or you end up needing to keep the treasury slider high just in order to afford your troops upkeep... which causes more inflation. Cue death spiral. Yearly income by contrast doesn't cause inflation, and as a rule its what you want to be subsisting off of.
One last thing before we move on; at the top of the budget window we can view our tech group. We're in the Oriental or Eastern Group (which is actually more or less Eastern Europe. Yes, it's stupidly named.) All countries in the game start divided into tech groups, which determine the rate at which they tech up (in addition to some other things). Tech groups run from Western (or Latin), which techs at 100%, to New World, which techs at 10% (or did in the last expansion. The numbers have changed a bit I think). The Eastern tech group techs at 85%, which isn't too shabby. Basically we'll out-tech anything to our east progressively as the game goes along. However the Muslim group (the next one down) start with a couple of tech's lead on us in all areas. This makes a pretty notable difference.
So that's the budget. Let's look at the military situation.

The top area shows what our preferred troop types are. These are the troops we recruit from provinces we own with our cores (more on cores a bit later). Better troops get unlocked progressively. Troop types are determined by what army type group you're in (top of the window again). These roughly match up to tech groups (and at the start of the game you'll tend to have the same in both). Eastern group troops early on aren't bad, but are notably inferior to Muslim group counterparts. Most of our neighbours are in the Muslim group, so we're fighting uphill in terms of troop quality in the early game.
Below the troop types is our forcelimits and military spending. Forcelimits represent the maximum number of regiments (or ships, for naval limits) that we can easily maintain. Note "easily". This isn't a hard limit like it is in most RTSs. For every percent over our forcelimits that we go, troop upkeep costs go up by a percent. So if we double our forcelimits, we double the upkeep costs for troops (in addition to having double the number of troops, meaning roughly 4x the cost). It's usually to be avoided except in times of desperation. Our forcelimits are a mixed bag. 4 Land force limits isn't great, although for a two province country it isn't terrible either. 9 naval limit though is very good for our size, allowing our fleet to punch above its weight. It's not enough to take on the Ottoman fleet (not by a long shot) or the Italian and Spanish mediterranean powers. but we can mess up most smaller navies in the region.
Next up, policies.

Policies are one of a few ways for you to customise your country a bit, alongside Decisions and Ideas (more on those two when they pop up). Policies are present as a number of sliders. We start the game able to move any one of those sliders one shift to either side. I don't want to bog this down by going over what every single slider does at both ends (I'll do so in a followup post if anyone's interested), so for now suffice to say that we're making a shift towards Free Subjects (raising army morale and reducing tech costs, but raising infantry and stability costs). Every time you move a slider, you will trigger one of three random events associated with it. In this case we've got a minor negative event from the Free Subjects move. A bit annoying but not the end of the world.
Next up, advisors.

At the game start you'll get a random selection of advisors. They're basically dudes you can hire to provide a boost to one thing, in return for paying some money to them on a monthly basis. None of these ones are particularly worth the cost, so we'll move on. Note that advisors are exclusively available to their home country for the space of one year, after which they'll be offered to any country in (I think) the same tech group. This also means we should check back in a years time and see what's available.
So, trade. Trade is a good way to get money for a small country, as it provides a fairly flat amount of income that doesn't increase with size.

Our provinces trade through the center of trade in Venice (although honestly that's more important for Venice than it is for us), and we're also part of the Venetian trade league, allowing us to trade in their Centers of Trade with Venices trade chances. Won't go into the details too much but basically trading through Venice is our best option. We prioritise it and send our starting merchants there.
Lastly, decisions and our starting mission.

Missions give bonuses for doing various things. There's a bunch of generic ones, plus some nation specific ones. This is one of the latter. It's also a bit ambitious at the moment (nick stuff off the Ottomans), so we'll leave it be. Below the mission are our National Decisions. These are another way of customising a nation. Most decisions give a bonus in exchange for a penalty, and do so in the long term (until game end). Some of them just give bonuses, but they're rare and usually nation specific. We've got one of those (Reestablish the Theme System) which will give us a permanent percentage boost to our manpower (which we need to build and reinforce troops). It also requires us to steal large chunks of the Ottoman Empire though, so it's another for the "things to do later" list.
So, basically we're small, weak, and have a massive green Ottoman menace in almost every direction. Do we have any advantages? Yes, yes we do. Firstly, said big green menace is currently busy fighting the Timurids to the east, who are even bigger and more menacing than the Ottomans are if anything. Second, we're an Empire. The Empire system of government is unusually good compared to other starting government types (Switzerland and Ming are basically the only countries with better ones off the bat), and it's normally a pain to get into, but we start with it due to our illustrious Byzantine heritage or whatever. Lastly, and most importantly, cores. Cores are one of three ways the game tracks who provinces belong to. To summarise, ownership (the primary one) determines who currently owns the province formally. Occupancy determines who controls that province in a war; a province can be owned by one country but occupied by another; occupying a country in EU3 doesn't immediately turn full control of the province to you like it does in a lot of other games. Cores are the final method, and they're a bit more complex. They more or less track whether your country has a legitimate right to a province. Not all provinces you own are necessarily cores, and by the same token, not all of your core provinces are necessarily ones you own. If another country owns your core provinces, you have an easier time getting them back and potentially incur no infamy from doing so (infamy tracks how much the world perceives you as a threat. High infamy is very bad). Cores that you don't own go away after 50 years unless the province is your culture, or you fight a war with the owner (which resets the countdown).
These are our cores.

Yeah that's right. We have a claim to just about everything in the region. Glee.
So, we're surrounded by enemies but we have claims on all their stuff. The obvious plan presents itself; find the smallest and weakest of those enemies and pick on them! That would be Candar.

When you declare war, you'll be requested to select a Casus Belli, basically an excuse for war. Different Casus Bellis allow you to demand certain peace terms easier and with reduced infamy and/or increased prestige. In this case we're taking Reconquest, as it'll let us take all of Candar with no infamy hit, because
it was totally ours to begin with before the bloody Turks came and ruined everything. It should be noted that picking any casus belli is preferable to not using one; declaring war without a casus belli incurs a negative two hit to stability (because your population wonder what the fuck you're doing) and an infamy penalty (because the world at large wonders the same thing), in addition to paying full costs for everything.
We've got a decent navy (as mentioned above) which can ship two regiments around. What we don't have is a decent general.

Generals can be recruited in exchange for money (which we don't have) and with skills based off of your Army Tradition score (which we don't have either). With that in mind we'll use the fallback option of giving our Emperor field command. Generals have four stats. Fire (boosting your armies effectiveness with guns, and consequentially absolutely useless at this point in the game), Shock (boosting your armies effectiveness with other weapons) and Maneuver (boosting your armies movement speed) all go up to 6. Siege (faster capture of forts) goes up to 3. We've got 1 in everything except Maneuver, which isn't too shabby.
With the Emperor leading them, our navy starts shipping our army over to Candar, and runs into a Candari fleet trying to do the same thing to us.

Sorry, did I say "fleet"? I meant "ship". This should be a walkover.
Karaman punishes me for tempting fate by intervening in the war on Candar's side.

This isn't too big an issue. Karaman are going to need a little while to get their act together, long enough to land in Candar.
We do just that, then send our fleet to go pick up another two regiments (our other starting one, plus some cavalry we just built) to go deal with Karaman, who have just decided to land a regiment in Morea.
Whilst that goes on, we move to engage a newly built Candari regiment in Sinope.

Candar's troops are better than ours one for one (being Muslims) and probably have a terrain bonus. But we've got two to one numerical superiority, and they've got low morale. Should win this.
Our troops arrive in Morea and deal a stinging defeat to Karaman, annihilating their attacking regiment.
We're also victorious in Sinope. I opt to split our army and send the cavalry to pursue Candar's regiment whilst our infantry besiege Sinope.

It's a bit of a gamble, as we no longer outnumber the Candari unit, but hopefully they've been beaten enough that they don't put up much of a fight.
Meanwhile our navy catches the Karamani fleet in the Aegean sea.

In this case it actually is a fleet, as they've brought two ships.
The Candari gamble doesn't pay off, and our cavalry retreat, battered a bit, to Sinope.
On the upside, Karaman's fleet is mercilessly crushed.

This more or less removes them as a threat for the time being, as they can't get troops over to attack us.
Buoyed with optimism by this success, we load the Morean army back onto our fleet and go have a look at Karaman's coastline, probing for a weakness.

A weakness that isn't particularly apparent. Karaman's remaining army is bigger than what we can ship over, making it hard for us to attack them. Still, at least things are going well in Candar.

...at least things
were going well in Candar. I've landed the Morean army in Sinope, however, so this should turn around soon.
Suddenly, an event!

The game has a lot of random events, which make you pick between a couple of different beneficial options or (more commonly, because the game hates you) a couple of harmful options. Technically this is one of the latter, but one of the options is a shift towards Aristocracy. Whilst Aristocracy is usually regarded as a weak option compared to it's counterpart, Plutocracy, it's still not too bad, and we're already pretty far over towards it anyway, so this is if anything a small bonus.
We capture Sinope!
And I almost forget to pause when October rolls around. As I said before, one year into the game you'll suddenly get access to a very large number of advisors. It's about a year in now.

Sadly I missed it by a few days so a lot of the good ones are gone, but we pick up a Master of the Mint.
Masters of the Mint provide a monthly reduction inflation. This can offset the monthly increase from the Treasury slider, so we can set that up a little without incurring inflation. Which is nice.
Regrouping our 4 regiments, we descend upon Candar's army with vengeance in our hearts.
We engage...

and defeat them. But we don't quite wipe them out.

Their battered army retreats into Sinope. This is affectionately referred to as pingpong by the Paradox forums, where an army will bounce around a few times between losing battles before finally collapsing. In some cases its a bloody nuisance but here it's pretty safe.
They bounce one more time from Sinope back to Kastamon.

Arriving in Kastamon however, their army is wiped out.
We consolidate our troops in Kastamon, to mop up their remaining fort.
As we occupy all their provinces, and they don't occupy anyone elses, we're able to annex them.

You can annex any country that meets the above criteria, so long as it's not too large. There isn't a hard and fast cutoff for what constitutes "too large".
If you're in a position to annex someone, they're completely beaten and there is nothing they can do about it.
You'll notice that our land force limits have gone up, from 4 to 6. That's the benefit of more territory.
At this point it occurs to me that Trebizond,
which is also totally ours by right, is just sitting there without any allies.
We gain Turkish as an accepted culture. This happens when a certain percentage of your owned core provinces are of another culture. It's a good thing.
With its ally crushed, Karaman tries to weasel out of its war.

The lesson for Karaman here is that begging for your life shouldn't involve trying to demand money off me.
Najd offers peace.

Oh did I forget to mention that? Yeah we're at war with Najd. And the Jalayarids. It may have slipped my mind, considering that
they're entirely landlocked countries that can't actually threaten us in any tangible way. Pests.
We occupy Trebizond.

Notice that we have only 99% warscore, and can't annex. That's because Trebizond are occupying a Timurid province. Not a worry though, for various reasons I'm not planning to annex them anyhow.
Instead, we'll vassalise them.

Vassalising a country makes them pay you a portion of their tax income, as well as giving you a boost to your forcelimits. Not as much as if you controlled them directly, in general, but as they're a one province minor they have a significant bonus to their tax income (and forcelimit, as the latter derives from the former) due to that province being their capital. If we annexed them, that would disappear. Additionally, they form a useful buffer against the Timurids. And as a vassal they'll automatically accept alliance requests and calls to arms, which means we can call their troops (troop?) into our wars if we want to. Finally, as they're part of our religion group we can diplomatically annex them later on if we want, and as they share our culture we won't lose our core on them.
As the Timurids are at war with our vassal, we're dragged into that war.

The Timurids are a horde country, which as of the last expansion are now permanently at war with their neighbours unless one side accepts to pay tribute to the other. This is why everyone in the region is taking the opportunity to beat them up.