Train Travel in Italy

Have the website of the Italian Railways as your planner (use Italian place names!). For the long-distance trips Rome-Florence and Florence-Venice Italo offers a competitive service. Look for the prices for a random date in - let's say - april and you get a good indication of what the tickets will cost you when bought in advance for September (possible from 90 to 120 days before departure date). You'll see that a rail pass is about the worst deal you can have.
A day trip from Rome to Pompeii involves a lot of travelling. Many advise to visit the ruins of Ostia Antica, just outside Rome.
Pisa can easily be done as a day trip from Florence. Regionale trains take around 1 hour. Saves a change of hotel.

You can easily explore train schedules for the major trips on the TrenItalia website. You'll need to use the Italian forms of the station names:

Roma (Tutte Le Stazione)
Napoli (Tutte Le Stazione)
Pompei
Firenze (Tutte Le Stazione)
Pisa (Tutte Le Statione)
Venezia S. Lucia

You can look at the departure times, trip durations and prices, then choose what looks like the best combination on the day you buy each ticket. For the longer trips (Rome to Florence, Florence or Pisa to Venice, and the Rome-Naples-Rome part of the Pompeii trip), you will save money by buying the tickets for the fast trains well in advance. It's too early for September prices, but I see that a lot of the prices are loaded for early June, so you can take a look at that period (using the appropriate day of the week to match your schedule). That shows how low the fare can be if you buy when tickets initially go on sale. Then look at prices for tomorrow or Monday to get an idea of what you'd pay for a last-minute purchase.

It's unlike that a rail pass would save you money.

I would not move from a hotel in Florence to a hotel in Pisa. The two cities are 50 to 80 minutes apart by train, and checking in and out of hotels is a big hassle. Most people find they need only a few hours in Pisa. Furthermore, spending the night in Pisa puts you farther from Venice; you'll just have to travel back through Florence anyway.

Your trip to Pompeii involves a change in Naples. Others can provide more details about your options there.

When I first started planning for our trip, I was a little confused about the different types of trains. I found it very helpful to go to Trenitalia's website and just put in a route I know I planned to take (i.e.: Venice to Manarola) and then see what results came up.

I ended up booking this combo ticket:
Venice-Florence: high-speed train
Florence-La Spezia: regionale train
La Spezia-Manarola: regionale train

Rome2rio.com is a great website to begin. Enter the locations (English or Italian names) and the info on the left column will tell you how long the travel time will be and by which modes of transportation. That column also has the link to Trenitalia where you can check departure times.

Stay in Florence one extra night and visit Pisa on a day trip from Florence. I would actually add nearby Lucca to the day trip because Pisa can be seen in one hour and is not as pretty as Lucca IMO.

What I think we want to find out is, if you buy tickets in advance, do they send paper tickets to your home in the USA or do you print out a paper with a confirmation number and then show it to somebody at a ticket window at the train station (like with airplane tickets), or do you enter your confirmation number into a ticket machine at the train station and then the machine gives you your ticket, or do you print your ticket from home, like with amtrak trains in the USA, or what is the procedure? How do you trust that it will work? How do you trust that they will keep your reservation?

Yep, what Roberto said. Don't stay in Pisa.

Mike, when you buy a rail ticket on the Trenitalia website, one of the options is a "print at home" ticket. they send you an e-mail with a PDF file as an attachment. You print out this PDF and this is what you need to bring with you. There is NO check before boarding, there is nothing to prevent you getting on the wrong train but your own common sense (but they are well signposted).
Sometime after the train has left the station the conductor will come round and check your ticket.

Example here: http://1.bp.blogspot.com/-rx9TTLjBmFs/U5bHT0Z-8PI/AAAAAAAAACw/IwovhL8FRBE/s1600/trenitalia6.png
Note it lists date, time, from, to, train number, coach and seat numbers.

The process with Trenitalia is simple and reliable. Payment by credit card. An emailed pdf for printing at home. (Which I always print two times to have one spare in case it gets lost somewhere.) Never had any problem with it.

Mike, the way they know you will use your reservation is that your credit card has already been charged for the full cost of the train trip.

ChrisF, there are now ticket checks before boarding at some major stations, at least some of the time.

Credit cards are not checked in Italy, as of August 2016, as they are in some other countries.