St-Paul-de-Vence is drop-dead gorgeous; think Taormina-level beauty, but smaller--truly minute. It was absolutely overrun with tourists during my 2017 visit. It would definitely not be my choice. It's a fairly significant bus ride (50 minutes) to get to Nice, which would be your lifeline to the outside world. Any time you have to take a bus to a different town to reach a train station, your options for convenient day-trips by public transportation shrink. A month is a long, long time in a tiny village.
The Fondation Maeght in St-Paul-de-Vence is definitely a must for lovers of modern art, but otherwise there seemed to be very little in the village other than tourist shops and touristy restaurants. And the visitor load means you most likely wouldn't be able to casually wander over to a local restaurant and get a table for lunch; I think you'd need a reservation or would have to wait. Mind you, the bus tours may all disappear before dinner time, for all I know. I had no interest in hanging around to find out. Did I mention that the place is tiny? Go to Google Maps, zoom in, and look at the key to see the distance across town.
The nearby town of Vence is larger and more real-world but is a bit farther from Nice (on the same bus line, so accessible from St-Paul-de-Vence); no train station there, either. Vence lacks SPdV's magnificient hillside setting but is definitely worth a visit or two if you're in the area. It has a very pretty historic district and I saw not one other tourist during my visit (during the afternoon siesta period). I liked Vence much more than SPdV, but tiny little touristy villages push all the wrong buttons for me.
Lignes d'Azur runs most of the buses in that part of France. You can look for schedules here. I consider most of the Riviera hill towns awfully isolated.
St-Remy is much larger than SPdV but it, too, lacks rail service. It has bus service to Arles and Avignon, making trips to those towns relatively easy. For most of the commonly-mentioned small-town destinations in Provence, I think you'd have to start with a bus trip to Arles or Avignon and connect to another bus or a train. Again, not really convenient for a bunch of day-trips if you don't have a car. But St-Remy is large enough that you probably wouldn't need to leave it all that often.
Gardens: The three I remember are all in the eastern part of the Riviera. Menton is the warmest place in France. It has two very nice hillside gardens that are worth day-tripping to. Menton also has a good-sized historic district, but it's a coastal town, so that area is hyper-touristy. None of those tourists seem to find their way up to the gardens. There are also some attractive public green spaces in Menton with flowers. That's not unique to Menton since the Riviera climate is so mild.
The Villa Ephrussi de Rothschild on Cap Ferrat has beautiful gardens, and the rich-person's villa is quite beautiful.
There's a garden in western Nice near the Asian Arts Museum. I didn't have time to visit it, but it was recommended to me by a local. My assumption is that it is not as great as the pay-to-enter spots in Menton, but that is just my guess.
Nice has at least 6 art museums. Many small Riviera and Provencal towns have art museums, often much better than you would expect to encounter outside major cities. I assume Rick's book that focuses solely on southern France does a good job of covering them, but I haven't checked. I'd recommend also skimming TripAdvisor's Things To Do listings to be sure you don't miss something you would enjoy. My two favorites in Nice are places no one else on the forum ever mentions: the Naïve Art Museum and the Asian Arts Museum.
The hill town of Biot is known for art glass (small museum and shops); the Leger Museum is on the bus route from Nice.