Orvieto, Assisi, Perugia and Spoleto

I would guess that buses may be better than trains for connecting some of these.

You can start with Rome2Rio to see what's available between some of your city pairs; if it's a bus, go to that bus company's website to investigate further. On Sundays, bus service can be greatly reduced or non-existent, so be sure you're looking for the day(s) you'll actually be making the trip.

https://www.rome2rio.com/

We picked up a car in Orvieto last year then spent 3 nights in Assisi. Drove to Spoleto, Spello, Perugia and a few more towns. We decided a car would be useful for this portion of trip. Then visited some amazing small towns near Orte where my great grandfather was born. Lots of great small towns in the area they call Tuscia. Highly recommend Calcata.

We stayed in Spello for five nights and visited Assisi, Perugia, and Spoleto and also took a day long wine tour. It was a delightful stay all via public trans (except the wine tour - private car picked us up).

While we loved tiny Spello and would stay again, some skill in Italian is useful and the train station is about a 20-minute walk from the town center. Perugia is larger and may make a better base.

Orvieto is a little off the track from Spello, Perugia, etc. Easy to visit it from Rome for a day, though.

It's all very doable by train, which is much more comfortable than a bus.
Orvieto is a nice place to stay, but can also easily be done as a day trip from Rome: only 1 hr.
Spoleto is 2 hrs from Orvieto and 1,5 from Rome. It is a good base for Assisi: only 30 mins. There are regular buses from the railway station to Assisi proper.
Perugia can also be done as a day trip from Spoleto: 1 hr. There are city buses from the railway station to the city centre.
For schedules and prices consult the website of the Italian railways. (Mind using »Roma« and not »Rome«).
Regional trains don't need advance booking. Buy the ticket at the railway station (at the window or from a machine).

We stay in Perugia at the nice Hotel Fortuna and make day trips by either bus or car. We took a bus to Assisi and it was very easy. We drove to Gubbio, which was very easy and the town is a really nice destination. They have a most unusual funicular you must hop on while it is moving.

Generally, we visit Orvieto as a day trip from Roma. It's a 75 minute train ride from Stazione Termini in Roma. We spent three nights in Orvieto once, which was two nights too many. Nice town, though.

Buon viaggio,

We stayed in Perugia and used public transportation, either train or bus, to visit Gubbio, Spoleto, and Assisi. The Tourist Info office in Perugia recommended best transportation option for each location. It was easy and we liked it all. Stayed at the Hotel Sant'Ercolano, reasonable, convenient location. Near the mini metro for fast access to trains and bus. Perugia itself wasn't hectic and busy and made a nice break in between day tripping. Have a great trip.

You are smart to choose one region if using public transportation. You can get to many places by train, more by bus (except on Sundays).

Assisi and Spello are a few kilometers apart, you could even walk to Spello and take the train back to Assisi. From Perugia, you could go to Montefalco; I would spend a night there.