Six days in Nice area

Will you be driving or relying on public transportation? If you have a car yes but if you're depending on taking the bus then no.

You can combine some of them. I would skip Monaco and Cannes. Our favorite day trip from Nice is Cap Ferat with village of St-Jean and Villa Ephrussi. You can walk everywhere including to Villa Kerylos in Beaulieu.
We combined Villefranche with Cap Ferat. Our favorites were Antibes and Cap Ferat.
If you go to Monaco give it only a half day. We did the Cathedral ( tombs of Princess Grace andPrince Rainier) and a guided tour of the Royal Palace. The city was mobbed with tour groups from cruise ships when we visited .
Go up to St-Paul de Vence and then continue on to Vence to see Matisse’s Chapel.
We love Nice and spend a lot of time exploring it. Our most recent trip ( our third) was for two weeks.

Why is this post in the "GREECE" forum?

Not sure why a previous reply said no. You can easily combine Monaco & Menton. And another day, combine Cannes & Antibes. The trains along the coast run about every half hour; otherwise, the bus is easy, too. I took the train from Nice to Menton, and took the bus from Menton to Monaco for a short stop. Then I took the bus back to Nice. Another day, I took the bus from Nice to Villa Rothschild - agree that it’s a wonderful place to tour. I really enjoyed the expansive gardens.

For Menton, be sure to walk through the gorgeous area around the Basilica of St. Michael Archangel where the buildings are colorful and the lanes are hilly.

If you don't want to spend a great deal of time in either place, I think you can combine Monaco and Eze. I believe the bus from Nice to Eze goes through Monte Carlo

You will most likely receive more responses if you post this on the France forum.

None of these places are in Greece :)