This is one expression where the context is important and implies different meanings. Sorry about the long answer. See Erica's answer, normally you say "I ride a bike". I would like to add a few notes:
Depending on the phrasing of your sentence. You would say "I took a ride on my bike" to mean the same thing, but you must have the "on" here;
"Ride on a bike" may mean being a passenger. You say to your little sister "Let's go to the shop, you can ride on my bike", the context here suggests you are the driver and your sister is the passenger.
You will often hear the expression "context is everything".