Train tickets


This is our first trip to Italy and we couldn't be more excited! This is for this year, 2020.
This is our itinerary:

3/27 - arrive in Rome by plane - stay until 3/30.
3/30 - train to Orvieto - stay 1 night
3/31 - train to Florence - stay until 4/4
4/4 - train to Venice - stay until 4-6
4/6 - fly home

My question is whether we should book our trains in advance or buy our train tickets as we go? We like to have an itinerary, but want to be flexible in our travel times.

Also, while in Florence, we'll go by train to Cinque Terra for a day excursion on 4/2. Again, should the tickets from Florence to La Spezia be booked in advance? or to any of the other towns there that we could book to?

Another day trip we're considering is to rent a car for 1 day to travel back through the Tuscany and maybe visit a winery and other towns along the way. Should the rental car be booked in advance?

I know these are alot of questions, but any help would be so appreciated!
Have you already booked your hotels or accommodations? I'll assume you have :)

Since most will have a time for you to be out of the hotel (generally 10am), it would make sense to plan for a train shortly after that gives you time to check out and arrive at the train station, without too much wasted time at the train station for you to just head to the platform and get on your train.

For that reason, it makes sense to go ahead and buy some of your tickets already.
I would recommend you get tickets for the fast trains for sure, the Fecciarossa or Frecciargento, because those are "numbered seating" and you want to be sure you have a spot on those trains. That's the case for your travel between Florence and Venice (and from Venice back to Rome, if you are flying back out of Rome).
The train ride between Rome and Orvieto and between Orvieto and Florence are on regional trains. The main difference here is that those are not "numbered seating" trains, they sell you tickets for the type of train and you get on and find any seat you find free that you want to sit at. So tickets for that train can actually be oversold and you can ride on crowded trains, with no seating at all.... can happen! In any case, these trains offer flexibility - you can just head to the train station when you're ready and buy the ticket to the next train that arrives at the station and get on that one ;-).

I don't think you need to book the train to La Spezia ahead, for the same reason - a regional train.

Car rentals - yes, definitely book that ahead of time! Especially if you want an automatic gear and not a stick shift!

Enjoy planning your trip, it is almost here!