Nice to Genoa to Milano

My sister and I are traveling to Nice, then Genoa, and then Milan at the start of July, and I was wondering if anyone has tips on what to do, places to eat, and reasonably priced but good hotels to recommend?
We are currently planning:
4 days in Nice
3 days in Genoa
2 days in Milan
From Nice we are considering day trips to Cannes, Èze, and Menton. Would you recommend those, or are there other places nearby we should prioritize instead?
We are also wondering how the train travel between the cities is. Is it easy/reliable to travel by train from Nice → Genoa → Milan, and should we book tickets in advance for early July?
Any tips, recommendations, or things to avoid would be really appreciated