jeff,
To begin with, you might find it helpful to attend the next meeting of the Sacramento RS group, which meets once a month at a local restaurant. Here's a link to their last meeting.....
https://community.ricksteves.com/travel-forum/general-europe/sacramento-travel-meeting-saturday-5-17-10-00-a-m
That would be a good way to "fine tune" your travel plans with an experienced and cheerful group.
Regarding your questions......
You can check each of your rail trips using the Trenitalia website, as that will indicate the type of trains that will be used on each route. One point to note is that schedules for August may not be uploaded yet, so just pick a day within the next week, using the same day-of-the-week that you'll be travelling (although the schedules seem to be working tonight for dates in August).
If you'll be using any of the "premium" trains such as the Freccia, InterCity or similar, you MUST have reservations which are specific to train, date and departure time. On those trains, you'll be assigned a Car No. (Carrozza) and Seat No's (Posti). If you miss the train listed on your ticket (or board the wrong train), you'll face hefty fines which will be collected on the spot! You can save money with pre-purchase of tickets for those trains, but note that you'll be locked into using ONLY the train listed on your ticket.
When travelling on Regionale trains, it's awkward to buy tickets in advance and there's no reason to do so as there's no price break. One important point to mention is that tickets for Regionale trains MUST be validated prior to boarding the train on the day of travel. Failure to do so will again result in hefty fines collected on the spot (these are often about €50 PP, and double if not paid at the time). Tickets have a "shelf life" once validated, so DON'T validate them on the day prior to travel.
If you're going to be buying P-P tickets in Italy, I agree with a previous suggestion to buy them at least the day prior to travel, either using a Kiosk or at a staffed ticket window. The Kiosks have an English-language option and are very easy to use. The ticket offices in larger stations often have LONG queues.
For the trip from Florence to Siena, you might consider travelling via Bus instead of train. The trip is only about an hour and you'll be dropped at Piazza Gramsci in the centre of town, rather than the bottom of the hill where the rail station is located. Again, you'll need to validate your ticket prior to boarding the Bus.
For the trip from Siena to Rome, you may travel first to Florence and then connect with the high speed Freccia train which travels at up to 300 kmH. That will depend on exactly which trains you choose. Florence to Rome is only about 90 minutes and it's a very pleasant (and quick) trip, so an efficient use of your travel time and somewhat of a "novel" experience. If you use that method, you'll have to take a Taxi from Siena down to the station (it's relatively cheap so that's not a big deal).
If using Buses or Metro in Rome, note that you'll also need to validate tickets there prior to boarding. For getting around Rome, I often buy a B.I.G. (one day) or B.T.I. (3 day) pass, which only has to be validated on the first use. The B.I.G. pass expires at midnight on the day of validation, while the B.T.I. pass expires at midnight on the third day after validation. They can be used for Buses, Metro, etc. within the central area of the city.
One final point to mention is that you'll have to be clear on which stations you're using in each location, as many cities have more than one. You might also do some research on the aspect of train changes, as you WILL be dealing with that at some point.
Buon Viaggio!