Southern France 9 days

There are plenty of places in Provence you can travel to without a car--more than enough to fill four days. It's the little places that could be awkward, time consuming or in some cases impossible to reach without a car, so the question is whether seeing some of the little places is important to you. (Of course, a one-day bus tour from one of the larger towns would be an option.) These places are well-served by public transportation, though you'd need to check to see which ones have the best connections so you end up with an efficient itinerary:

Aix-en-Provence
Avignon
Arles
St-Remy (bus from either Avignon or Arles)
Nimes
L'Isle-sur-la-Sorgue
Marseille
Cassis

I'm not clear on the logistics of getting to the Pont du Gard without a car.

Agree with acraven that there are many places you can visit using public transportation in the area. I would add Eze, Menton, Monoco (the country), Antibes, and Cassis to the list. It’s getting out into the Luberon that a car is really beneficial.

Thanks a lot acraven and rachele for your replies. What is the best home base to visit Arles, St Remy and Nimes and Pont du Gord?

I'd say not St-Remy since it has only bus service. Nice is a bit far west (actually just outside Provence), so from that list Arles seems best. However, I'm not an expert on Provence, having only spent a week there, staying in Avignon, and I didn't get to Nimes or the Pont du Gard.

I trust some folks with more Provencal experience than I have will provide additional input.

Do try to visit Pont de Gard if you go to Provence.
For the Provence part , a car will give you freedom to visit the smaller towns. Honestly, I can’t imagine touring Provence without a car. Avignon and Arles may be visited by trains but I’m not sure as we had a car.
From Nice , which is in the Côte d’Azur not Provence, you can easily do day trips in that area by bus and train along the coast. Antibes, Villefranche, St-Jean Cap Ferrat, Monaco, Eze and more. You can even go into Italy.
Rather than trying to visit “southern France” which is a very large area , why not stay in Nice and visit the Côte d’Azur easily by train and bus as Nice is a transit hub. And Nice is wonderful! We spent two wonderful weeks in Nice and did day trips for two weeks on our third visit there. .
Or rent a car and visit Provence. But Provence needs more than a week to see it fully.

There are plenty of half day trips from Nice to consider (Antibes/Biot as well as Vence /St. Paul de Vence could pair up as two day trips), plus Monaco could be a day trip and Eze a half day. You can use the train or bus for all these. Sightseeing in Nice itself also can be managed in ca. 3 days.

Will will be going to the same areas from 4/27-5/10. We are renting a car, but can certainly understand not using one for the Riviera. However, we do need the car for Provence.

Compared to Nice, Arles - Nîmes and the Pont du Gard are practically not in Provence but closer to the Languedoc Roussillon region. Culturally and socially and historically it's a different region. Even the local accent is different. It has nothing to do with the Riviera

If you want to visit this area, you will have to stay in the region of Nimes or Arles much further West than Nice.

With a car Uzès is also a place where you can stay.

https://www.france-voyage.com/tourism/uzes-1420.htm

Eventually Montpellier which is the main city of the region.

Public transport is not really suitable for tourist visits in this region. Given the few days you have it would be better to rent a car
However you can go to the Pont du Gard by bus from Ales (1 hour) or Nîmes (45 minutes) or Uzès (15 minutes)

We will be in the Nice area for a week late April. We are basing at Beaulieu Sur Mer (and Nice for a night before we fly home) as we are not renting a car and using trains and buses. I’d encourage you to get the RS Nice and French Riviera Snapshot as he explains the villages and public transportation well. We plan to see: Monaco, Antibes, Villafranch-Sur-Mer, Cape Ferrat (possibly Menton) and Nice. We will visit Nines for a day trip while in Montpelier.
We visited the Provence towns on a previous visit.

Thank you so very much for all of your replies. We are still deciding if we want to rent a car and drive to Avignon/Aix and explore that area for 4 days or just explore Nice and the surrounding area for the entire stay.

8 full days around Nice would be a bit much for me, personally: splitting the time between the Nice area and areas further west makes a lot of sense.
The area around Arles/Avignon is the "obvious" choice, but you could also choose to stay closer to Nice and explore the Verdon Gorge and the beautiful villages to the south of it (Tourtour, Cotignac, Bargemon...). You'll be two-three weeks early for lavender blooms around Valensole but you never know, the year could be unusual.