Ohh I'm so jealous!
As a PP said, breakfast in Campo de Fiori is a must, there is a little cafe with quirky chairs, tables and tiny sofas in bright colours, great for people watching. Check the narrow streets off it, most of them are named after the artisans who once had shops there, Via dei Capellari (hat makers), Chiavari (keys) etc.
Don't be scared of the long Vatican queue, I've been 3 times and it never took more than 25 minutes to get in. Definitely don't buy "fast-track" tickers from the dodgy sellers swarming around the place, they book you on a tour which is every 45 min or so, so you'll end up waiting even longer.
If you are into art there are a couple more galleries (aside from Doria Pamphilj) not far from Fontana di Trevi, Galleria Sciarra and Galleria Barberini.
Piazza Navona - yes it's quite touristy but again amazing for people watching. Look for Piazza del Fico which is a few streets away and Bar del Fico, they both take their name from the fig tree in the square. There's a nice mix of independent shops, impossibly trendy young italians and old men playing chess with a cup of espresso.
The Pantheon is gorgeous.
Jewish Ghetto is great for lunch up and down Via del Portico d'Ottavia, if again a bit touristy - there's lots of trattorias famous for their artichokes but check Nonna Betta, Hosteria da Giggetto and Piperno. For amazing history look for Teatro Marcello and Portico d'Ottavia; for a smile look for Fontana delle Tartarughe in Piazza Matei.
Trastevere is great for night life, but not too raucous - Piazza di San Cosimato and Piazza di San Calisto were the most lively spots. Most trattorias serve the same sort of food so you can't go wrong.
Not far from Trastevere you'll find the Palazzo/Galleria Corsini, amazing collections including Caravaggio, Fra' Angelico and Titian.
If you have a soft spot for Art Nouveau, check Quartiere Coppede. It's fully residential, no bars or restaurants, but the most eclectic architectural mix you can think of - Baroque, Greek, Gothic, there's even a chandelier at the entrance.
For street art, go to Pigneto or Ostiense. Also in Ostiense you will find Eataly, huge food hall if you want to stock up on gorgeous Italian food.
For hip Rome, Testaccio is the place to look for. There is a contemporary art gallery called the Mattatoio, housed in what it used to be a slaughterhouse. Opposite there's the Testaccio market with lots of great options for lunch, both stalls and sit down places. If you're feeling brave try Mordi e Vai, delicious sandwiches with unusual ingredients (tripe, kidneys etc). The owner is a living legend.
For lovely, romantic Rome go to Piazza della Madonna in Monti. It looks just like a film set. There are some beautiful churches in Monti too, San Pietro (with a statue of Moses by Michelangelo) or Capella din San Zenone for the gold mosaics.
I have saved the best for last - Garbatella, my favourite place in the world, a gorgeous maze of courtyards, gardens and fountains. It's a working class neighbourhood of rust and copper coloured buildings, with a sort of village feel but also an old Rome charm hard to put into words.