Debbie,
SNCF (French national rail) and Trenitalia (Italian national rail) are currently having a dispute over shared services. Because of that the timetables on their sites are incomplete for cross-border services.
The best timetable site for virtually all of Europe is the German national rail site, www.bahn.de. Click on the drop-down flag menu for English. After you get summary timetables for La Spezia-Nice or Nice-Milano, click on the arrow to the left of each departure time to see the connecting points, if any. You will also see the type of train for each leg and whether or not a reservation is required for that leg.
If a reservation is required, it is included in the price of the ticket. Seats are not likely to be sold out on the Italian trains you will take so you could buy tickets literally at the last minute, but it's a good idea to buy them a day or two ahead of time.
No reservations are possible on regional trains in either France or Italy, including the ones you would take between Nice and Ventimiglia.