You're going to Rome, caput mundi.
Here are some tips:
1. There is one of the best place on earth for pizza in rome. It's called pizzarium, the owner, Gabriele Bonci, is very famous in Italy as he's a master in his art. It's a great take-away (the last time I went there there weren't many seats and it was full of people), good selected ingredients, and also the beer is great.
2. Ice cream is good everywhere in Italy, but one of my favorite one is Grom: they have 2 shops in Rome. Not as many flavours as other places, but every month there is a "Flavor of the month" and the whipped cream is real whipped cream, i.e. not coming from a spray bottle.
3. If you want to taste real Rome food, go for the Jewish ghetto, as people lived there since Romans time and through all the ages. If you go in the area of Via del Portico d'Ottavia you will find a lot of Kosher restaurants, patisserie and bakeries...
4. Catacombs? http://www.catacombe.roma.it/en/index.php are a bit out of the way. There is also a necropolis below St. Peter's but belongs to the Vatican and I'm not sure how and where you can book a visit.
There is so much to see in Rome that I don't know where to begin, but I think that St. Peter's is a must, it truly takes your breath away.
And a stroll in Via Vittorio Veneto at sunset or in the evening, just like in La Dolce Vita. And trastevere, too..