First, get Rick Steves Provence And The French Riviera. This has almost all the information you need, including very detailed logistics about getting from place to place. His general France book has much less information about this area.
You only need one base for a hotel, as you can see all of these places by bus or train from a single base. Nice is the most central with the most connections, but it's a city. If you prefer a smaller place, Rick likes Antibes (west of Nice) and Villefranche (east of Nice).
I will say that I disagreed with many of Rick's opinions about the various places. I liked Monaco and Cannes more than he did, and Villefranche less. And he barely mentioned Juan-les-Pins, which I liked. I also really liked Haut-des-Cagnes, where Renoir's house is. So, don't be afraid to see anything you're curious about, and form your own impressions.
Again, the book will explain the transit details, but along the coast (Cannes to Antibes to Nice to Monaco to Menton) trains run twice an hour until about 11 PM, and buses 3-4 times an hour until about 8 PM. The buses are dirt cheap to encourage people not to drive, and the trains are only a bit more but are twice as fast; both work well, although for Cannes I found the buses took forever and I much preferred the train.
Buses to inland destinations are less frequent, but certainly work. For instance, there are 8 buses from Nice to Eze Village on Sundays, and 16 on other days. A hard one to get to without a car is La Turbie, with only 4 buses a day from Nice and 6 a day from Monaco (and none at all on Sundays). The other hard one is St. Tropez - it's not near the others and has no train station, so you take a bus or boat. But you won't run out of places to see. In addition to the ones you listed, and the two I mentioned in the last paragraph, there's St. Paul de Vence (the Fondation Maeght is a very interesting art and sculpture museum), Grasse, Villa Ephrussi, Villa Kerylos, and more.