I assume you are relying on public transit.
Day 2 is no problem. If you decide to lunch in Eze, make a reservation in advance. VSM is pretty small and there are few small sights like the Cocteau chapel, the Citadel and Rue Obscure. The beach is quite popular. If you are walkers, there is a walk up Mont Boron to Fort Alban (near Elton John's villa, nearby at the entrance to the area you can catch a bus down to Nice or walk down the other side). You can also walk around around Cap Ferrat. There is also a steep walk down from Eze but good shoes are required and it is much more of an effort.
Day 3 you can hit the market early. Just be aware on Monday it changes to a brocante/antique market. The Villa Ephrussi is beautiful, as are the gardens. Some people also choose to walk around the peninsula (per above) or spend time in St Jean. It is also easy to hit the Villa Kerylos in Beaulieu-sur-Mer as the bus does a little loop in center of the town (and then people sometimes take the train back).
Day 4. You can hit Monaco rather quickly or if you are into gardens and aquariums, or a formal meal, you can spend a day. It is pretty easy to spend a half day there and continue on to Menton (or vice versa). Alternatively, you can go to Mala Plage, which is one of the priciest private beaches on this section of the Cote d'Azur. There actually is a nice village at Roquebrune but the high village is separated from the lower train station by a distance/climb (as such, it gets less tourist traffic ).
Day 5: in Cagnes-sur-Mer, there is a second part, called Haut de Cagnes. There is a shuttle bus up to the perched village and castle. If you are coming from Nice, you probably know there is a transfer to the bus to St. Paul de Vence, You can visit the Fondation Maeght in SPdV and/or continue on to Vence (Matisse Chapel, nice old town). There are some regional buses beyond this point but the frequency and ease drops dramatically. By car, it is easy to continue on to Tourettes-sur-Loup and Gourdon, two beautiful villages. (BTW, Grasse is even further away)
Day 6: Ventimiglia is most popular and best on market days (Friday). It is a bit grittier than the other towns along the Italian Riviera. Dolceacqua is a beautiful stone village, however. You can take a bus or taxi/uber there from Ventimiglia or if you have a car, continue on to Apricale. You can also go further into italy, to places like Bordighera.
Unless you spend time on a beach or have a great desire to get a pic on the faux red carpet in front of the festival area, Cannes has less unique things to offer, with the exception of the ferries that leave the town. Cannes and Antibes can be easily combined together , even on a single return train ticket.
But it also comes down to what are you interested to see? It seems that museums and beaches are low on your priority, and that towns and architecture are your focus?