Lots of threads with suggestions for clothes, so I'll focus on the food. Hard to go wrong just about anywhere there, but…
For low cost and amazing pasta, this is my mainstay. Pangies
. Small place but really friendly owner and delicious food.
For a quick sandwich and glass of wine, this is the highly popular I Due Fratellini
. Walk-up window, blazing fast service, and good company in the street.
For a family-style trattoria and dining with strangers (love it), try Sostanza
. The pasta is outstanding but their butter chicken is equally good. And they have a "house cake" that's made by a local baker, and it's probably the best dessert I've ever had.
If you are with family, have a car and a few hours to spend outside the city (it's about an hour drive out), I highly recommend La Cantinetta di Rignana
. Read the reviews
and, well, just go there.