The Greeks did as well, during the Battle of Delium for example, the Boeotian league fielded over 1000 cavalrymen. While the mainstray of their forces were infantry, they did use cavalry as auxiliaries.

The Macedonians of Philip and Alexander the Great used their heavy companion cavalry to great effect as well.

I can't speak for the Greeks, but the Romans did ride horses into battle. Horses made a quick way to scout out positions and deliver messages. In battle, they were indeed ridden. They had the obvious height advantage.

In the early periods of Rome, the first cavalry was supposedly created by the kings of Rome to provide protection. It was a unit of 300 horsemen, and increased until the unit was 1800 strong. The cavalry used to only include patricians, but it was later opened up to anyone who could afford a horse and gear. The calvary, in the Roman army, were usually divided and put on the outside of the foot soldiers. The horsemen were mainly support though, the main fighting force were the foot soldiers. The cavalry could be used to defend the main infantry or to help flank an enemies position or to prevent the legion from being flanked.

Here is one account by Appian of Alexandria talking about a mounted cavalry. This is referring to the Punic Wars. There isn't too much here to really dig into the uses of horses in an actual battle, but it does metnion that there were 30,000 footsoldiers to 3000 horsemen int his specific instance.

In the later Imperial Army, cavalry became more important and were better equipped for front line battles. These units were known as "Cataphractii".

Cant say much about Romans, who as far as I know used many mounted auxilliary units and later equites cataphractarii modeled after Seleucid and Parthian heavy lancers.

When it comes to Greeks, they used mounted units in many forms. Most city states and leagues maintained some amount of light cavalry for scouting and harassing, but there were also many examples of pretty heavy cavalry units used directly - Boetians were famous for substantial amound of heavy cavalry and so were Thessalians(who became famous for being the other strong cavalry force of Alexander the Great) and Macedonians. Later successor armies also fielded good amount of cavalry, especially the more eastern ones(Seleucid empire, Bactrians) who in some way had to counter eastern tribes and rising Parthians.

Not linking any sources cause Im on my mobile phone, but it should be pretty easy to find.