Doug,

With respect to going to the Vatican please note that the Vatican closes early on Saturday( I think last admission might be at noon) and it's closed on Sundays( except last Sunday of the month). If you can visit the Borghese Gallery, which is open on Tuesday- Sunday. You should book in advance. It has an impressive sculpture collection by Bernini, Canova and others and also a painting gallery.

Adrenn is right about Palma being one of the best gelato.

We were in Rome this past March and enjoyed L' Orso 80 in Via del Orso and La Campana in Vicolo della Campana, both near Piazza Navona( the latter claims to be Rome's oldest restaurant). They're not fancy but serve genuine italian food and are filled by locals.
Have a great time in Rome!