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Public transportation in Umbria is better than many parts of Italy because it is served by two branches of the FS state run rail line (trenitalia) and by a private rail line ( ferrovia centrale umbra). However, many of the stations in this hilly part of the country are a long way from the town centre and there are not always shuttle busses to take you to the town. You can find out which towns are well served and which have their station miles away by looking at this map drawn by Bill Thayer, in his Umbrian diaries.
Coaches also provide regular service to many of the towns. Unfortunately there is no unified web site showing all the different services so you'll have to go to the individual companies: Sulga bus, APM bus or SENA bus and look up their routes and timetables. Note that they are definitely cheaper than the trains, but they can be much less comfortable and may not be very good at keeping to their timetables.
Now, knowing about the public transport, all you need to do is choose which city to make your headquarters. Bill Thayer (the chap with the incredibly useful train map) recommends Trevi. Perugia is also a good choice: the city itself is fascinating -- you'll want at least 4 days to explore it all -- and it has a minimetro, outdoor
escalators and city busses which take you from the town centre to the
bus and train stations with ease so you can explore more of Umbria.