Towns/Cities as homebase on the Côte d'Azur
 

I'm planning to spend about 1 week in the south of France in June. So far, I have 2 nights in Provence (although not sure where exactly to stay), 2 nights in St. Tropez, and 3 in Antibes on the itinerary. Does anyone have advice about what the best "home base" towns should be for exploring the area? I want to see a beautiful, relaxing countryside in Provence, the parties and beach in St. Tropez, Cannes, Nice, Vence, Antibes, and Monte Carlo....

It looks as if your itinerary is planned for the 7 nights, so it is confusing as to what you mean by a home base.
You can spend 2 nights in AIX then go to Nice for 5 nights. From NICE it is very easy to go to all of the towns that you
mentioned by public transportation. A car is absolutely not necessary. The only exception would be ST.TROPEZ. You can take a day cruise to ST.TROPEZ that would give you about 4 hours to walk around and have lunch. Of course if you want to party in ST.TROPEZ this plan is not good for you.
If you have a car then you can spend 2 nights in ST.TROPEZ and then drive to NICE for the last 3 nights
Personally, I think that you are spending too much time going and coming. Skip something and spend more time in 1 or 2 towns