If we visit without a car, is there public transport to surrounding towns and villages? And are taxis very expensive?
Bastia Umbra has a railway station so you could visit towns along the rail line such as Perugia, Assisi, Spoleto etc. You could also take a train to Perugia which is a transport hub with some bus service and, for example, take a bus to Gubbio. But if you want to see the countryside a vehicle is the best option. I do not know the cost of a taxi.
Thanks so much for your reply; maybe we'll hire a car for a few days as well as using the trains.
One FUN way of getting around in Italy, and hence also in Umbria, is on a scooter ! There are scooter rentals available in Bastia Umbra (among other places).
If that is not your thing, I'd recommend renting a SMALL car..... the roads, towns and parkings are NOT large around here :-)