Horse Archers are a pain to fight because they use skirmish attacks the whole time and they move very fast. But they also take a long time to kill other units as opposed to Melee cavalry.
I would like to point out an advantage (among several others) that horse archers have over melee cavalry. While chasing a routing or retreating unit, or any other unit that turns its back to your HA unit, you will be delivering missile fire to the rear of the enemy unit where they don't get the benefit of any shield bonus and subjects them to morale penalties.

Equites are faster than your post-Marian heavy cavalry and will give you some pursuit capability.
I did this all the time when I played Roman factions
Another thing the others have not mentioned (I think) is that Armenia gets cataphract archers, which are simply awesome.
These are my favorites! Chariot Archer killers, excellent ranged attack, and can go over to melee when necessary because of good A/D melee stats. They're almost obscene, really

There are plenty of good cavalry factions, so it really depends on what culture you like, and who you like to fight.
The best advice of all!