How to travel the French Riviera with a 9mo baby and no car?

I need some tips for the Riviera. I've been to Nice and Cannes solo, but it's been a while. Now I have a baby and going there but will not be renting a car. What towns would you suggest we stay for easy access to enjoying the area?

I don't mind smaller ones, Nice is nice but would like suggestions of area to stay there that has a more local feel and easy to walk around with strollers.

All in all not interested in the glitzy Riviera experience, just want a decent area to hang out and eat on a reasonable budget.