Thanks for de-lurking! Hope you stick around and contribute, this place is great. (Though you shouldn't use me as an example - I've basically just started back up yesterday after a six month hiatus!

)
The fastest way to learn is to post session reports! Then the experienced members can chime in and offer some specific "this is how I would do it in that situation" advice.
That being said, the answers to your post would take pages and pages to fill. Each Faction has a couple of obvious weaknesses, and some more subtle weaknesses, and usually any of them can be turned into a strength.
As an example: Lizardmen. Really there is not much ranged attack power there - only a couple of minor units, and most of the time you won't find army lists here with any archers in them. You just rush across the field and eat people!

But then every once in a while... someone pulls out the Swarmling Bowmen and goes running around flanking and shooting just because the opponent doesn't expect it. Or you toss in a mercenary like Elementalists behind Tyrant Spearmen and suddenly you have a very real stand and shoot threat.
The beauty of the system is that you have the capability to convert perceived weakness into strength - it's a whole new meta-game once you've mastered the basics.
I would highly recommend reading the in-depth Faction tactics discussions (at least the first few pages) that are stickied here. But more than
Faction weaknesses, focus on the general principles of Battleground that are true across the board:
1) Pinching is king!
2) You're trying to win your good matchups before you lose your bad ones
3) Make a hole in the enemy line and pinch him to oblivion!
4) Ranged attacks support your melee guys, they typically don't win on their own
And keep at it! You'll find it suddenly clicks and then it opens up a whole new world of tactics.