Denny's right. This is the sort of thing local tourist offices are usually extremely helpful with. They'll often be able to hand you a printed-out bus or train schedule. It's true that it's a lot easier to find reliable online train schedules than bus schedules. I never fully trust bus schedules until I've seen them posted at the bus station or asked at the ticket window (where English will probably not be spoken). Information from the tourist office is not quite as comforting to me as information posted in the bus station, but it is close.
For short day-trips I don't find much difference in the convenience of buses and trains. It's all about the departure time, how long the trip takes, and where the station is located at each end of the trip. A direct train will usually be faster than a bus, but sometimes the train isn't direct. Buses are usually cheaper.
One thing about buses in Italy is that you usually need to have a ticket in your hand before you get on the bus. For that reason I prefer to get on the bus at the bus station itself (if the town is large enough to have one, as I assume Perugia does). Otherwise, I assume I'll have to wander around, looking for a Tabacchi shop that sells bus tickets. I'm quite certain there are exceptions to that, but I'm unclear on the rules and where they apply.
An easy way to check on train service (existence and schedule) to any particular town is to use th Deutsche Bahn website. It won't give you fares on Italian trains, however. And you won't know without looking at Google Maps whether the train station at the destination is convenient to the historic area you probably want to see or is a mile or more away at the bottom of a very steep hill.