Planning on going to Provence for a week . Is there a recommended base city that will be easy to travel in and out?
Nice or Villefranche-sur-Mer
Those cities work well for exploring the Riviera, but not for visiting what a lot of us think of as Provence. Rick recommends Avignon or Arles as bases for Provence and discusses the differences between them in his book. I think Avignon has somewhat better overall transportation connections, but it matters a lot what specific places you want to go.
While Nice is in the region called "Provence-Alpes-Côte d'Azur", most people would indeed not refer to it as being in Provence. The informal limit is west of Cannes and Grasse, roughly.
So, assuming you are referring to the region around Avignon, Aix, Marseille, etc.:
Avignon is the most straightforward base regarding transportation. Easy to reach Arles, Orange, l'Isle-sur-la-Sorgue, Aix (bus), Pont du Gard & Uzès (bus), St-Rémy (bus), Nîmes, Marseille from there.
Arles is smaller, arguably more charming, but connections are a bit worse. You lose easy access to St Rémy, Aix, Pont du Gard, Uzès.
Aix has poor train connections, but good buses. A poster here recently spent a month there without a car (CaliMom I think). There are good links to Avignon, Marseille, l'Isle sur la Sorgue, Lourmarin, La Ciotat
I stayed in Arles this past spring and just loved it. IMO it is more attractive and charming than Avignon. Although Avignon does have better transportation connections, you can get to Avignon quickly by train from Arles. They run frequently, but it does add extra time to the trip.
Balso described the Avignon/Arles conundrum perfectly. Having been to both, I’d stay in Arles over Avignon in a heartbeat. I would rather deal with the compounding travel difficulties (you may have to take a short train to Avignon from Arles to get further afield) than stay in Avignon. Avignon offered an array of half/full day minivan tours, and these can be a great way to see more of Provence without a car. Direct train connections are limited to towns in the region. Public bus connections can vary a lot depending on time of year. You will need to be flexible.
Still, a week in either of these towns is too much imho and I would suggest a few days in another town. Vaison-la-Romaine is an all time favorite for me, with enough to occupy you for a couple days. Whatever you do, try hard to be in any town for their market day.