Hello lplarkin401, and welcome to the boards,
In broad terms there are fast trains and local trains called regionale. Regionale are all the same class, don't sell out and there is no reason to buy those tickets until you need them. My wife and I call these "Get your butt on the train" class tickets.
There some exceptions like InterCity trains which also have specific tickets, but in general these are the big towo.
In general sites will book you through but especially going from large to small towns or vice versa you will need to change trains. This is fairly painless and straightforward but booking direct trains is always easier if they are available and the timing works for you.
If you check Vernazza to Lucca you'll see a very short local ride to La Spezia, then La Spezia to Lucca requires a change in Viareggio so this is less than straightforward but the are all Regionale so for one price and one ticket you can get on board and after a couple of changes and 2 hours and bit you'll be in Lucca.
Lucca to Florence is a but more straightforward but will require a change in Pisa. Check for trips with one change rather than two and choose the time that works for you.
Florence (Firenze S. M. Novella) to Rome (Roma Termini) you will want to book a fast train - either Trenitalia Frecciarossa or the private Italotreno aka .Italio. Fast trains come with classes, seat reservations and specific times and are cheaper the earlier you book them.
Resellers are okay for checking schedules but the actual train company sites are here:
https://www.trenitalia.com/en.html
https://www.italotreno.it/en
Note: You will have to use the Italian names for some cities like Firenze on the Italian version of the site.
Get the Trenitalia app to check schedules, track trains and buy tickets through the app.
Great advice and information on trains in general here from the man in seat 61: https://www.seat61.com/Italy.htm
Hope that helps, it seems like a lot but don't let it throw you but remember thousands of people do it everyday,
Have a great trip,
=Tod