I have a bulged disc and some other problems. I started with a hybrid (I think they call them fitness bikes now), but now I ride a road bike. Go to your local bike shop and tell them about your back problem and price range. They can suggest several bikes. Having the proper frame size is the most important if you have a sensitive back.

Just start slowly and work your way up to longer rides. Cycling actually improved my back problems by strengthening my core. Be sure to do some extra stretches for glutes, quads, and hips.