Should I base myself in one place or move around Italy?

Fewer bases, longer stays — almost always. Every move costs you half a day and a night of comfort, and unpacking once is worth more than one extra town. For two weeks, three bases is a good rhythm: a city, a countryside stay, and a second city, with day trips radiating out from each. Trains make regional day trips genuinely easy, so you can sleep in Florence and still see Siena and Bologna. Where basing fails is the Amalfi Coast and Sicily, where distances look small on the map and take three times longer than you'd expect on the road.