Skirmish isnt hard at all. Infact the skirmish A.I. is pretty stupid. At first I was like 'WTF this isnt fair? WTF!?! How the fuck? Oh come on!', but now it's pretty boring because I win every time.
Personally I always play as Allies, I havent play axis much yet in the demo. But my strategy is simple. I, personally, like to keep a mobile force that is adaptable. So I tend to favor infantry and light vehicles over armor units. IMO the allied armour units kinda sucks against the axis armour (which is historicly correct IIRC).
The computer love to ninja capture points, which is why I always have a jeep or M8 as a rapid responce unit to prevent that. Instead of having 1 base 9 miles away from the action, I try to have a secondary base right in the middle of the map, that ways I can reinforce my units in much less time, or call in additional forces without having to wait for them to run half way across the country.
The AT gun is a freaking god send. This unit can rounds after rounds from a tank, turn around, and make the tank its bitch. Unfortunately this unit is pretty stupid and it wont turn around to fire back without you telling it to do so, and even after that the tank can easily backup and flank the AT crew. Because of that it's essential to have a rifle/ranger/paratrooper squad with AT ability. close to your AT.
And upgrade your riflemen! The riflemen is a tier 1,2 and 3 unit. They can mow down other infantry units, pin em down, take out a tank, and they only cost 270 manpower.
And use weapon support! Heavy machine gunners and snipers will muderate any soft target stupid enough to get close, and motar men is useful in taking out AT crew and machine gunners, while providing close range artilery support.
As for the 3 doctrines.....it's tough. The infantry doctrine provides a fairly powerful infantry unit with good anti AT capabilites, and its howitzer is damn powerful. Airborne provides paratroopers and para AT, providing instant reinforment against infantry AND armor. Armor gives you the ability to capture points with light vehicles, allows you get a powerful rocket tank and big big tank.
They're all good, it just a matter of using whats goes with your strategy the best.