How difficult is train travel in Italy?

You can obtain train tickets online on the Trenitalia website. It's better because you block the seats.

There is certainly a high-speed connection between Florence and Rome. It's called Freccia Rossa. It takes 1.5 hours and starts every 30 minutes.

The 22nd I would do Florence-Pisa-Florence-Rome.

Florence-Pisa takes more or less one hour and you have trains every 30 minutes.

Venezia-Florence is also a high speed connection (Freccia Argento). It takes two hours and stars every 30 minutes.

Edit

As @user1187008 said, a good alternative to the Pisa-Florence-Rome route is the Freccia Bianca train. Direct Pisa-Rome connection that takes 2h50m and costs less than the Freccia Rossa

In general, with high speed trains between the major tourist cities (Rome, Florence, Venice, etc), you are better off booking in advance, especially for the high season summer months. The cheapest seats tend to sell out before the travel date, so you may end up paying more than you'd like.

As mentioned before, Trenitalia is the longest established company for fast train travel between major cities, but you should really check out their new competitor Italo.

As they are newer, Italo has brand new cars, a newer website, and tend to sell out less quickly. I tried them in summer of 2013 and was very pleased with the service. Trains were on time and easy to deal with, conductors were very helpful and spoke English. They also have "lounges" in certain stations where you can buy tickets and talk to customer reps. I'm sure it's only a matter of time before this secret is out and they will sell out at the same rate as Tren Italia.

One note: both web sites have quirks for booking, although Italo had far fewer. You might need to try a few times or try a few different payment methods to get through.

Long distance or high speed trains are very easy to use, while the regional trains may be less user-friendly. All high-speed trains fares go up as departure approaches and seats and fares seem to just about equally sell out across companies (a few hours before departures in many cases).

There are three competing companies operating high speed / long distance trains in Italy:

• Trenitalia have the most trains. Their online ticketing is very easy, you just get an email and find your seat on the train. Their customer service has gotten very good since Italo began competing with them.

I guess your idea of buying tickets on the spot is to have some flexibility, to that end the Eurail Italy one country pass (US$362 for all of you) is valid on Trenitalia. With the pass you would still need to pay for a seat reservation (10 Euro each) for every train. You could plan the trains you intend to take and secure reservations on them early (to make sure you've got a spot if they sell out), but if you did miss the train you'd only be out the 10 Euro and could be asked to be placed on the next train with available space (again for 10 Euro) you might have to wait a few hours for space, but at least you wouldn't be out an extra full ticket price of a few hundred Euros.

• Italo has a decent high-speed product and runs on the lines your interested in pretty frequently.

• ÖBB Italia (Austrian Rail Italy) isn't currently on the routes you're looking for.

There is a lot of great advice here. But to answer your question, you can walk in and buy the tickets. You can purchase the tickets either through the automated ticket machines or at a counter. Probably a good idea to see when the trains run so you wont spend hours waiting at the station.