Train journey from La Spezia to Nice

The train system is very easy. Go to the Trenitalia website (in English) and compare the options. Genoa is a large station and 8 minutes may be difficult for you. Ventimiglia is small and you will need less time to transfer. There may even be an option with only one transfer which also would be simpler for you. Rome2Rio is good for getting an idea of transportation options but they are not always accurate and I certainly wouldn’t buy any tickets through the site.
You will regret having a lot of luggage, however, every step of the way.

Rome2rio is a reseller that combines multiple ways of travel and companies into a single site. It's useful to learn how you can go from A to B, not to buy tickets since they do not run anything.

The Italian system is quite simple when you buy tickets on line: Find your train on trenitalia.com, decide if you want savings or flexibility and buy a ticket. At the station, find your train's number on the bilingual boards headed "Partenze/ Departures" and walk to your platform. If you get on the wrong car, wait for everybody to be seated and walk along the train till you reach the right car.

It reminds me how we all find our flights around the world, the only difference is that the 2 most distant "gates" in the biggest italian station are 400 meters away and that trains never sell out.

It would be different if you decided to buy tickets at the counter right before getting on, but the online system is idiot proof.

You can also go on Google Maps and go from x to b. Then click on the icon for mass transit. It tells you the train schedule hour by hour. Often you have to mix in trains and buses--but not on this journey.

I use this method all the time to figure out how to travel from town to town.

I sent you a PM with a suggestion for getting more info.