Suggestions for a Travel Base Between Nice (France) and Milan (Italy)

Look at staying around Turin (Torino). There are a lot of sights to see around there, and the roads go in all directions from there. You can easily get to Milan--fast.

Most people are visiting the big three--Rome, Florence & Venice. But Turin is also a very good place to visit as much of the industrialization of Italy is in the north. The mountains north of there are also something--just south of Switzerland.

Since you have been to Turin, and bearing in ming your need for public transport, then maybe Genoa if you do not mind a gritty city (Genoa is fascinating behind the grit)? Or somewhere on the coast to the east, like Camogli or Santa Maria Ligure. Plenty of day trip options, including Cinque Terre, Portofino, Genoa.

We have greatly enjoyed staying 5 nights each in Genoa and Camogli in November 2018, and in Chiavari last September --- each is very close to the other two by train, so which one to pick would depend on what you are looking for. If I had to pick one for you, I'd say Camogli --- it's a very pretty little seaside town. You will eat extremely well, especially if you like pesto and focaccia.