I'm not 100% sure, because I haven't managed to get to Gubbio yet, but I think you'll find it a slog from Assisi. Gubbio doesn't have rail service. Bus E001 from Perugia takes 75 minutes and doesn't run all that often. Although service is rather frequent between 11:05 AM and 2:10 PM, there's a gap in departures between 8:10 AM and 11:05. You'd need to pay careful attention to the schedule in both directions.
There's a 7:42 AM train from Assisi that arrives in Perugia at 8:02 AM. That's way too close for comfort for me. The previous train departs at 6:49 AM. On a vacation, I'm not getting up before 6 AM to catch a train to a bus. And I don't like the idea of reaching my day trip destination at 12:17 PM, which is what happens if you take the 11:05 AM bus from Perugia.
It doesn't appear to work any better if you take two buses (E007 + E001), changing in Perugia.
Buses to and from Perugia
The best I can tell, Perugia is the transportation hub for that area. With Gubbio in the mix, I think Perugia is a much more practical base than Assisi. When you have no car, you need to consider practicality unless you have a lot of slack in your schedule. On the other hand, Spello and Spoleto do not appear to be a problem from Assisi--faster than from Perugia. It's really Gubbio that's the issue.
Do pay attention to the train as well as bus schedules in this area as you plot your itinerary. They look somewhat spotty.