Without knowing specific dates and arrival/departure times for each city, it is really tough to comment further.
Catching a public transportation bus to catch a train is probably not going to happen.
Livorno to Genoa is 180km or about 2 hours drive. Livorno to Nice is 370km or about 4 hours drive. Rent-a-car and drive. Probably then only way to go.
Good luck!
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