Jim,
Use the Trenitalia app to choose the type of train and departure time for each of your trips. It will specify what type of train is being used for each departure. If it's an Intercity or Freccia, those have compulsory seat reservations which are provided with the ticket. The ticket will specify the carrozza (coach number) and posti (seat numbers) where you must sit. Tickets with reservations are specific to train, date and departure time and can only be used on the train specified on the ticket. If you board any other train, even by mistake, you could face hefty fines which will be collected on the spot!
If you're using Regionale trains, there are no reservations possible, and no price breaks for buying advance tickets. Buying tickets from an automated Kiosk in the station is very easy, as they have English language menus and take different forms of payment (that's what I often do). However one important thing to remember is that paper tickets MUST be validated (time and date stamped) in the small machines near the tracks.. Failure to do so can again result in hefty fines collected on the spot. Regionale tickets bought online don't have to be validated that way. This should explain the process - https://www.youtube.com/watch?v=cvfgKGm5ZjQ .
Tickets for the Metro and Buses in Rome and other places usually also have to be validated. I often just get a multi-day pass which only has to be validated on the first use, but I haven't checked that lately so you'll have to confirm.
It's likely that you may have one or two changes on some of the routes you mentioned. Use the electronic sign boards in the stations to locate the Binario (track numbers) of each train you'll be taking. You'll probably have to use the underground passageways to move from one track to another.
The station at Assisi is at the bottom of the hill, so you'll have to get the bus to get up to the town. Check the current Rick Steves Italy guidebook for details on that. When you arrive in Orvieto, you'll have to take the small Funicular up to the town.