From: Utlima Thule
just to add to the above.
It is complex, I've been playing since release and still find I've been missing/misunderstanding something.
The good thing is to some extent you can situate yourself in that complexity as suits you.
At one end you can fully automate the air war, your role becomes that of the strategic manager who tells 8 AAF to concentrate on tank factories this week and your tactical airforces to prioritise bombing rail lines over close ground support. At the other end of that scale, you can determine which bombers hit which factories using which particular payload.
The ground war benefits from relatively low density. Even in the main campaigns by say Oct 1944 you probably are only dealing with 50-70 divisions and not all these need to be moved every turn.
I'd say the best way in is via the scenarios. The game suggests start with Sicily - I personally wouldn't. Even if you stick to the pre-game set up you have to handle air borne operations, naval assets as well as quite a complex little puzzle on the ground.
My suggestion would be start with the breakout scenario. Its all set up for the Cobra offensive, it'll teach you a lot about ground combat and the intersection of ground and tactical air. Then either one of the later 1944 scenarios where you will start to have to pay much more attention to logistics or (if you have Torch) one of the N African scenarios.
These are gems, low density but unit usage and placement really matters. And a pretty good crash course in how the logistics game works.
As you progress, the 1943 campaign is perfectly playable in a month or so vs the AI, esp if you either automate the air war or know what you are doing. If you get to early 1944 and find its all gone wrong, its no huge loss to step away and restart (or go back a number of turns). Most people have had to do this, as the linkages between 1943 (Italy and strategic bombing) and 1944 (France and tactical bombing) are something that you tease out with practice.
There is a lot of good resources both in the AAR section and the War Room (the tips and tricks thread is very good)