Ride a bicycle. This is a particular usage of "ride" which implies that one is also "driving" the bicycle.
You would say "ride on a bicycle" only if the person was riding the bicycle and someone else was driving, which might happen if someone was sitting behind the driver, or sitting on the handlebars, for example. Also, if you change the phrasing, "ride on" can become correct, like this: "He took a ride on his bicycle."
Best Answer
One rides in a chariot.
Sources:
Man in a Chariot Episode
Wikipedia
2 Kings 9, and 10:16
Thanks @josh61 for this link showing that while "riding on a chariot" has been used, it's always been much less common: Google Books Trends