My parents spend the winters in downtown Melbourne and they love it. I can't speak to bike trails, but downtown Melbourne is quaint, on a small harbor, and has a stretch of cute shops and restaurants. Its also just over the bridge to the ocean and beach if you change your mind on that. They also like to visit New Smyrna Beach which has a fun strip of restaurants and stores, but is right on the beach. If you're willing to go further, St. Augustine is great too.
ETA: My recommendations are all on the east coast. I agree with PP below that the gulf has some nice spots without doing the beach, but I'm not sure I would describe them as "quaint" and they are >1 hour drive with traffic. We've always find the gulf to be much busier and touristy compared to the laidback, easier to navigate Atlantic side. Pros and cons to both.