What town in Umbria would be best to stay in if I want to take day trips to other towns via bus or train? I don't want to drive.
Looks like Perugia is your best bet. Frequent trains to Assisi, Spoleto, Passignano sul Trasimeno, Spello. Buses to Gubbio etc.
Perugia is a transportation hub for Umbria (trains and buses). I never drive in Umbria or Tuscany (although sometimes I travel with friends who drive), and you can see plenty without driving. Travel within a region is usually easier to do than traveling from one region to another using buses or regionale trains. When you have a few places you want to visit, you can check the Trenitalia site for travel times, connections, costs - just put in a date within 7 days of the date you use the site (even if it's nowhere near the actual date you'll be traveling), since regionale train schedules aren't usually listed very far in advance. For buses, the SITA line still serves most of the region. For reference once you get to Umbria, a local bus is a "bus", a long-distance or intercity bus is usually called a "pullman".