Is this sentence correct?
'This piece makes me want to get on my bike and ride off/away.'
Grammatically there is nothing wrong with it, although without context it seems a little odd in meaning!
While "ride off" does mean much the same as "ride away" I would probably avoid it, and give preference to "ride away", in a sentence in which I'd already used the preposition "on", unless I was creating a deliberate effect. "Ride away" is certainly more common - See here.