tb,
On the routes that you mentioned, you'll likely be travelling on Regionale trains, so there's absolutely no reason to purchase these in advance. You will not have any difficulty getting return trains each day (unless of course there's a strike happening). You can easily buy the tickets at the station in Florence, either from a Kiosk or from the Biglietteria (ticket office) there (often long queues at the ticket office, so I just use the Kiosks). One IMPORTANT point to mention is that you MUST validate those tickets prior to boarding each train on the day of travel. Failure to do so may result in hefty fines, which will be collected on the spot!
I checked an arbitrary date next week, and it appears all the trains from Firenze SMN station to Lucca are all Regionale. Some are direct and some have one change at Pisa Centrale. I'd suggest taking a direct train if possible, as the travel time is considerably shorter.
The trains from Lucca to Pisa Centrale also appear to be all Regionale, and the travel time is about 30 minutes. As I recall, there's also a Bus on that route. Again, DON'T forget to validate your tickets (that applies to the Bus as well).
Some questions - are you planning to visit Pisa just to see the Tower and then return to Florence the same day? Also, will you have luggage? If you don't have luggage, the Pisa San Rossore station is closer to the Tower but there's no luggage storage there. If you want to store your luggage, you'll have to use Pisa Centrale.
For the return trip from Rome to Orvieto, there are a variety of trains on that route, including Regionale and InterCity. Have a look at the Trenitalia website (if it's working) or the Bahn.de website to see the solutions. You can pre-purchase tickets for the InterCity, as long as you're willing to commit to a specific train, date and departure time, as the tickets can ONLY be used on the one train specified on the ticket. If used on any other train, you'll be subject to hefty fines which will be collected on the spot! If the departure you choose is a Regionale, just buy tickets at Roma Termini the day prior to travel, and then validate before boarding the train. DO NOT validate Regionale tickets the day prior to travel, as they have a "shelf life" of about six hours once validated (depending on the route), so will have expired by the time they're used (which of course will probably mean hefty fines).