The standout day trip from Rzeszów is Lancut, fifteen minutes east by car or train, where the Potocki castle is one of the great aristocratic residences in Poland. A morning at the castle and an afternoon walking the surrounding English-style park is a comfortable day out.
For a wilder option, the Bieszczady Mountains start about two hours south, with the Solina reservoir and the high meadows of the Połonina Wetlińska. Przemyśl, the old Galician garrison town an hour east, is also worth a half-day for its fortress and the painted Orthodox church at Posada Rybotycka.