don't know what sort of places you're after - casual or more formal, but here are some of my faves in that part of the region:
Terni: underrated town, with a wonderful wine bar: L'Oste della Mal'Ora. run by Renzo who's a top sommelier so wonderful wine selection, including by the glass. Simple, but superb food - great cold plates (eg goose or salt cod etc) and cheese selection. Don't know how good your Italian is, and Renzo has no English, but clientele is well-educated and very friendly, so somebody will be able to help out if need be.
Ferentillo: few km east of Terni is the Abbazia San Pietro in Valle, which is lovely and has a great restaurant just outside its door: Il Cantico. Absolutely beautiful setting and delicious food, including a great bread selection (always worth a note in Umbria!). Chef Renato speaks good English.
Going west from where you are, Orvieto is well worth a visit, and I'd presume would be on your itinerary. Couple of places really stand out there: L'Asino d'Oro, which is very well known and has a lovely outside eating area, so if you're booking request an outside table, and L'Antica Trattoria dell'Orso which I'd recommend making a booking; good hearty, but not heavy fare.
Finally, if you're after something a bit different, but a huge amount of fun and you'll be north in Spoleto (again not far from where you are) I'd head for Osteria del Matto, run by Filippo with Mamma doing all the cooking. 'Matto' means madman and Filippo will give you a great time with a varying choice of music, serious but good value wine selection and carefully sourced ham, cheese etc. Mamma will serve up all sorts of goodies from the day's market - and it's very cheap. We were there in April, and from memory it was Eur18 for dinner (there's no menu, you take what you're given, once you've explained if there's anything you don't eat) comprising 7 or 8 courses - each one on the small side but even so. Wine extra of course.
Apologies for slow response; have only just seen this. I know a couple of wonderful places in Foligno, if you're going up to the centre of the region? have fun!