Telefonini > Cell phones in Italy are called “telefonini,” the affectionate diminutive of the word for phone. It should be an affectionate word as people sometimes seem to love their telefonini more than their relatives. Anyway, I wish I could give advice on which one is the best. My first year here we measured this by how far from the windows of our apartment we could be and still have coverage, but this changes year to year. Kate says Wind* means you have to be outside in a breeze to get reception! Try the friendly (English-speaking) Luca at the Vodafone store in Piazza Danti behind the Duomo near the fountain.
Mediaworld > A massive electronamagic store which is at the Collestrada mall.
*Wind is a very common mobile carrier among Italian and foreign students