We flew into Nice last April, and decided to stay in Beaulieu-sur-Mer for our first 2 nights. Our itinerary was a loop, as our flights were round trip to/from Nice, but we were also exploring Provence. So we planned to stay in Nice at the end of our trip to make it easier to fly home.
I chose Beaulieu-sur-Mer because we were seeking a lovely and quiet place to get over jet lag, and we wanted to visit Èze one day, and the Villa Ephrussi de Rothschild and Cap Ferrat the next day.
Then we took the train to Arles for a few days, with a day trip safari of the Camargue. Next was Aix-en-Provence for a few days, including a wonderful full day tour of the Luberon. That included Lourmarin, Roussillon, Gordes, and Fontaine-de-Vaucluse. Then we headed back to Nice for a few nights. Nice was fairly crowded in mid April, and I predict it would be even more so in late June. One day we went to Villefranche-sur-Mer for lunch and some exploring, but it was raining a lot, so I don't think we saw it at its best.
Beaulieu-sur-Mer, however, was truly a "beautiful spot on the sea" as its name says. It's a very short train ride from Nice, and it was recommended to me by a forum member. We stayed at Hotel Select, booked on Booking.com, and it was a lovely place to stay. Not far from the train station, and we didn't see very many tourists there. It was easy to get to Èze le Village and the Villa Ephrussi from there.
For Èze I'd recommend taking the earliest bus you can take. We bought Lignes d'Azur refillable cards from a Tabac store, and were able to jump on the bus quickly in Beaulieu, and exit Èze quickly when the bus came and the crowds were all trying to figure out how to leave later in the afternoon. I can only imagine how busy it would be in the peak season. One special thing we did there was have a little treat at Chateau Eza. They were full for lunch and dinner reservations, but we could go and sit on the terrace mid morning and enjoy desserts and beverages with a spectacular view. We spontaneously decided to do that when I saw there were tables...if we'd waited until after our tour of the Jardin Exotique, we would not have found a table free.
We had a delicious lunch at Lavoglia in Nice, and especially enjoyed a giant creme caramel for dessert. They are known for their huge desserts - the two of us only ate half of the portion, and took the rest as carry out because it seemed a shame to waste it! I had it for breakfast for the next two days! :-) The teenaged boys next to us made light work of a giant chocolate eclair, and a huge portion of mousse au chocolat! That was such a funny surprise to see the huge desserts there. We'd go back there again!
So, I'd recommend considering Beaulieu if you seek a quieter place to stay. It is a short train or bus ride to get to Nice, so you could day trip, and retreat to a lovely spot afterward.
Have a wonderful trip!
Laurie