I have played these games for 5 years, starting with Combat Mission: Beyond Overlord (CMBO), and now into CMBB and CMAK. All are worthwhile to play, but there is a big jump in realism for CMBB and CMAK. BB covers the eastern front and AK is Africa and most of the western front. The good thing about these games is that while turn based, they use simultaneuous action for each side. You give orders to your troops, and then the game decides what happens in the next minute of play as your units try to do what you tell them. It evolves like a movie where you can give new orders every minute. No I-go, You-go crap here.
There are several competative ladders around. I am in a club that has alot of tournaments and campaigns. For the campaigns (usually) a operational- or strategic-level boardgame will be used, but instead of rolling dice to resolve combat, two players will fight it out in the Combat Mission game.
M38, playing other people is very easy and much more fun than the AI. For play by email (PBEM), just start a game as usual, but select PBEM where it asks which type. When you are done with the turn, it will ask for a filename to save. It generates a text file with that name in the PBEM directory. Email that txt file to your opponent. When you get a txt file, save it in the PBEM directory and open it from the Multiplayer menu option. It uses 3 emails per turn so that you don't see the results of your turn right after you plot your orders. This prevents cheaters from rerunning the turn multiple times to perfect their moves.
Live play by TCP/IP is also supported, both players will be plotting moves at the same time, then the turn will be generated and watched. You can set time limits for the orders phase if you want, so it becomes like a timed chess match.