Rivera

If you won't have a car, Eze will be a frustrating place to stay, as it has buses to Monaco (none on Sundays), buses to Nice (half as many on Sundays), and buses to Eze bord de Mer (daily, where you can catch trains and buses to other places). And that's it. And looking at schedules, they stop relatively early. So, you'll spend a lot more time than you want looking at and working around bus schedules, lots of time connecting to other buses to get to your final destinations, and/or a lot of money on taxis (particularly to take you back home in the evenings if you miss the last bus).

Nice is the transit hub of the region, and you can get anywhere from there. It's definitely the easiest place to stay without a car. If you want a smaller town, Villefranche sur Mer works almost as well (I wasn't taken with it, but most are). Places on the coast, rather than above it like Eze, have 4 buses an hour and two trains an hour connecting them. Buses to other destinations are less frequent, and tend to be very Nice-focused, meaning it's easy to get there from Nice, but from other starting points you may have to change buses.

Rick Steves Provence And The French Riviera has all the information you need about transit in the area (lots of great details). Note that his general France book has much less information for this region.

Page 368 of Rick's Provence book summarizes direct public bus options from Nice airport to Nice, Antibes, Cannes, and Monaco; bus to Villefranche requires a connection in Nice at Place Garibaldi. See also airport links.

Forget EZE. It's a one stop Wow. Villefranche sur Mer is where I recently stayed and I loved it. But I had a car. I think I would choose Nice if I did not have a car. More to do and see, easy to the airport and better connections to the region.