Ordesa y Monte Perdido National Park is the obvious day trip from Aínsa — and the main reason most visitors are here at all. The park's four valleys each offer different terrain and different levels of commitment. The Añisclo canyon and Pineta valley are both accessible from Aínsa without going through Torla, which means less driving and far fewer people in summer. The main Ordesa valley, north of Broto, is the most spectacular and the most crowded; go on a weekday in June or September if you can.
The high Pyrenees villages north of Aínsa — Broto, Fiscal, Sarvisé — are worth an hour or two each. Broto is a typical Aragonese mountain village with a Romanesque church and a small square; Sarvisé, a few kilometres further up, is even quieter and has views of Monte Perdido on clear days. The road between them, the A-138, is one of the better drives in the Pyrenees.
South of Aínsa, the Collegata de Alquézar is about 45 km away — a stunning Romanesque village above a limestone canyon with its own good hiking. If you're making the drive back toward Huesca or Zaragoza, it's worth a stop.