Without a car, explore the local trains to nearby towns from Nice as well as the longer distance trains that are less oriented toward daily commuters. There is very good service to major towns along the coast that make it easy to see museums, etc. There are also some ferry and hydrofoil services in the area, for example between Antibes and St Tropez. With a longer distance express train, possibly including the TGV, you could go to Marseille for a long day.

I don't know about the scenic train route that someone suggested, but I suspect that it follows the Napoleanic Route road for cars, which is indeed stunning although it can be a slow and tedious drive.

Some like to go to Grasse, an inland city where perfumes are made, but I was not impressed.