Which is correct, "ride on a bicycle" or "ride a bicycle"?
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."