I also stumbled onto this game rather "accidentally" a few months ago. I have bad a ball since then.
I think the "play vanilla" then play DLC is good advice. Both play differently; be aware briefings and prestige play differently between the two. There is a post somewhere in the stickies on that point (wish I had found it before playing all the DLC/GCs!). If I find it I will come back and link it.
But a visceral, battlefield level comment I have is to study the unit characteristics. Until you can look at a unit you have as well as an enemy unit and know it's relative hard/soft combat abilities, movement radius IN VARIOUS TERRAIN/WEATHER, zone of control effects, etc., keep right-clicking and studying. Also use "esc" to pull up the library and look at the terrain and movement effects on different classes. There is a ton of interesting, accurate subtlety built into the game. Studying it and learning it is a big part of the fun. Frankly, once you know all this, destroying the AI is easy at the hardest level rather quickly. But still fun.
Another thing I find immensely fun is to basically link the game to Wikipedia (not literally, but figuratively). I can be playing a river crossing in GC '43 and I'll go look up the river, the town, etc. in Ukraine or NE Russia. Read about it, history geography, etc. Or just go read about the Demyansk pocket before/during playing that scenario. Fascinating. And of course, equipment - reading the real subtleties of the MkIV variants or Panther, etc. is interesting (to me at least!). Fun to link the game to facts on line and see how well the developers/designers captured things (pretty impressive to me, frankly).
I get a big kick out of this as I was a history student a long time before taking a different career path. Doing these things, to me, links the game to reality and makes a very interesting experience. So to me, "getting the most from the game" involves more than which path you take in playing, but exploring the vast wealth of equipment, geography, historical chronology, etc. that has been replicated. Really impressive to me (I'm not a very experienced computer gamer. I've played StarCraft years ago, and the BBC simplistic "battlefield Academy" online - that was how I found this game circuitously). That's about it in the last 12 years. But this is a ton of fun.
Having said that, what I found "fun" was playing the 1st Poland scenario several times. I played it through on the Col. level twice, found it manageable. Set it at "Field Marshall" and played it until I could get a decisive consistently (six or eight tries?). Then I played everything on FM from then on. I really found that satisfying; I did not wonder if there was something harder or I was "taking it easy." And honestly, I think a person approaching the game seriously can get the hang of FM play quickly.
I have not played multiplayer yet; I'm sure that will be a new, humbling experience.
Edit: Ah, here it is:
the sticky from Kerensky about DLC vs. main game:
viewtopic.php?f=121&t=29134
the main game teaches you to finish in a time constraint to advance - dv ms. mv dictate success and camaign path. the DLC is not that way. You generally have ALL the prestige you need, and a dv vs. an mv is never really important. So a big lesson is to understand this and to play the GC games differently than the core game. I had trouble adapting for a while.
