For passerines in particular, small islands off Italy can be truly fantastic in spring. The tiny island of Ventotene (2 hour ferry ride from Formia, which itself is a 90 minute train ride from Rome or Naples) can be absolutely swarming with birds, especially trans-Saharian migrants, as they make their first landfall after crossing the Mediterranean Sea. A ringing station has been operating there in spring for nearly 30 years now and many rarities have been trapped, including some pretty astonishing ones like Cyprus Warbler.
For raptors, the large island of Pantelleria between Sicily and Tunisia can be excellent in April and May, with thousands of Honey Buzzards, and is the best place in Italy for eagles (Imperial, Steppe, and Great Spotted could be seen together there in May 2013). Some good passerine fallouts, too, though not as frequently as Ventotene.African Blue Tit and Laughing Dove are resident.
Linosa, the very best rarity spot in all of Italy, gets very little coverage in spring but is also very good (Sooty Falcon last year in May). Migrant falls aren't quite as frequent as on Ventotene, but they can be pretty impressive.
Ventotene and Linosa in particular have very limited breeding avifauna, so the vast majority of birds one sees are migrants.