Cavalry was not the Greeks' strong suit until the time of Alexander. The cavalryman of the classical period might cynically be described as a guy rich enough to own a horse and thus get out of the rigorous training that a middle class hoplite would get. In battle, they mostly chased down enemies who were broken and fleeing, or disrupted low-ranking troops like peltasts or psiloi. They guarded the flanks of the hoplites somehow, I guess by charging light and maneuverable troops when they tried to flank the phalanx.
Lacking stirrups and saddlehorns, the Greeks didn't do mounted archery or couched lance charges. They did have a nasty spear called a kalyx, which they would use with an overhead downward thrust, aiming for the neck or behind the collarbone. They also used the makhaira (Heeeeeeey makhaira), a single edged saber, shaped to deliver a slicing or chopping blow with the full force of a charging horse behind it. They'd also carry a few javelins, becoming missile troops that could be quickly deployed to a strategic point.
A lot of them went unarmored, but I prefer my figures to flash some bronze, so the ones I just bought have helmets and shields. They were marketed as Carthaginian cavalry by Gorgon miniatures but with their crested helmets and round shields, they read as very Greek.
Work in progress shots:
I also had some old horsemen sitting around. I probably painted these guys a year ago. It was encouraging to put them side by side and note my own progress.
Progress! Now I have to channel this positive feeling into repainting these older guys, which is not my favorite task.