in general, you need to try to shoot down any unit first, and then engage in melee combat... with cavalry, you need to attack the enemy from behind and also protect your own while the infantry fights... and in melee combat, swordsmen, axemen, etc. are better; they are good against spearmen, and against similar units... archers should be supported by cavalry, and spears should be used against cavalry =)... you already know that =)
P.S. Egypt is also overpowered... they always take the south... their infantry is strong even at the beginning of the game... their cavalry with axes is also good at the beginning, and even in later periods... also, their archers and spearmen are good =)).
Hmm... maybe I'll play Rome again, edit the document, and play as Denmark... although they all look the same, and after Medieval 2, it's already... meh... although the battles are more interesting there... I mean in Rome =)