A popular but worthwhile site are the dramatic Cliffs of Moher. They're only 50 miles northwest of Shannon right on the coast. The views of the Atlantic Ocean can't be beat. (Well, they can, as there are more dramatic cliff views near the Ceide Fields in north Mayo, but that's quite a drive north.)
The great medieval town in Ireland is Ennis, which is only 20 miles north of Shannon. Ennis is an evocative town that feels like stepping back 500 years. Kilkenny is excellent, too, but it is a quite a bit farther southeast.
Bunratty Castle & Folk Park (https://www.bunrattycastle.ie/) is only 15 minutes away from Shannon. It's a bit corny, but I'd go back again.