Use the pause key (press P). R:TW is faster than previous games in the series and it's only reasonable to expect that new players have difficulty adjusting to the speed. I've played both Shogun and Medieval before and still use pause anyway as I'm not terribly fast and like to have fine control over my units. There are more than a few veteran players here (mostly those who do multiplayer) who don't need to use pause/don't use it much, but IMHO it's better to use pause first and hone your tactics in deciding what actually works for new players.

The scissors/rock/paper analogy is appropriate for describing general counters to various troops:
Counter slow moving infantry with archers and ranged troops
Counter ranged troops by attacking them with fast moving cavalry
Counter fast cavalry with spear infantry

Play on Normal battle difficulty if you are still having trouble. That way no side gets attack bonuses and the stats for each unit can be interpreted as is.

I'm no battle expert but if you could give an example of a particular battle with details of units on both sides and general terrain, I'm sure the vets here could help you out further.