Tour of Umbria or use the train?

Renting a car to see the villages of Umbria would be easiest and best. Spello, Bevagna and Montefalco. Some can be accessed by train.

I haven't yet visited Spoleto, but have gone to Assisi twice by train (take taxi from the train station; there are usually a couple parked out front of the station that seem to plan to meet the trains; it's about a 6-minute taxi drive up to town). Orvieto is super-easy to visit by train, and I've done so three times: the funicular to the old town is directly across the street from the train station and runs approximately every ten minutes; a little bus is right outside the funicular station up top and will swing you up to the Duomo (center piazza) in five minutes. Perugia is on my plan for this December, and research shows me that there are both a bus near the train station and a "mini-metro" (small single cars on a mono-rail) a very short walk from the train station that will bring you up to the oldest part of the city. Looks like the mini-metro runs at least every five minutes. Spello is also on the line going to Assisi, and about 12 minutes by train from Assisi. I usually print out train schedules from trenitalia.com on-line before I go, for my own use at the ticket kiosks, or to use when buying tickets from an agent at a train station (some of whom, especially in smaller stations, do not speak English well). Many of the trains to these towns will be "regionale" trains that don't take reservations, but some trains that stop at Orvieto (on the line from Rome to Florence) will be Freccia trains that do require reservations. Have a great time!!

I've visited both Assisi and Orvieto by train, easy peasy. Lovely towns.

We stayed several nights in Perugia and visited Assisi,Spoleto and Gubbio all by either train or bus. The TI was very helpful in suggesting day trip options and the best transportation mode for each destination. Travel times are short so you get a good solid day for your money. It was easy and we enjoyed it. Safe travels.

Since you haven't been there before, you'll want to concentrate on the cities you mentioned rather than the tiny villages. Thus, train is fine. (And we have visited all of the places you mentioned by train.) You should definitely include at least a half day in Spello, and it's on the train line as someone mentioned. However, if you want to go there from Assisi, the bus is easier. If you haven't yet visited Arezzo, it's near Umbria and is another great place to visit.