We will be travelling by car in the Umbria region. Is San Giovanni Rotondo easily accessible from there? How long and difficult is the drive from Perugia?

It's not a difficult drive but it isn't a daytrip. We drove from Mattinata (near SGR) to Torgiano (near Perugia) and with a stop for lunch it took from 8 am until about 5 pm.

Did you stay overnight in or near SGR? If so can you recommend a spot?

We stayed for several days outside Mattinata at a resort called Baia delle Zagare. From there, we were able to visit all of the Gargano Peninsula.

We are primarily interested in visiting the places that honor St. Padre Pio. Is it worth the trip would you say?

Whether you drive or take the train, San Giovanni Rotondo is a long trip from Umbria. You will need to spend the night and the hotels at SGR are extremely basic, well worn places catering to pilgrims on a budget. Seeing the church and the hospital and even a bit of the town (there isn't much and it's jammed with tacky souvenir stalls) doesn't take all day. I just think that traveling all that way from Perugia may not be worth it unless it has been a life long dream of yours to make this pilgrimage to Padre Pio. We spent a morning in SGR, but we were based close by.