Our rule of thumb when visiting Italy is towns/cities in the winter and coast/countryside in the warmer months. Umbria and Northern Lazio are our favourite countryside destinations. A base in Perugia would be an ideal mixture of town and countryside for that time of year.
We really enjoy visiting Perugia (nothing to do with the chocolate!), the old city on top of it's rock is fascinating. More ancient Etruscan, Roman and medieval buildings etc than you can shake a stick at, they seem to be round every corner. We find it a very laid back city and because of it's altitude not at all claustrophobic. Though it will be cold as is the rest of the area, you might even see some snow.
The views over the surrounding countryside are wonderful, on a clear day you can see right up the Vale of Umbria which is lined by hilltop Umbrian towns, including Assisi, all waiting to be visited.
http://www.umbriatourism.com/
If you have the time, a car and weather permitting a drive in the Valnerina area, just south of Spoleto, is very rewarding in terms of scenery and ancient fortified villages.