The gelato is awesome. It's almost all excellent, but Il Gelato di San Crispino (Piazza di Trevi) or Giolitti (near Pantheon) are among the best.
These are the don't miss sights:
Pantheon
The Roman Forum
The Palatine Hill
The Colosseum
The Vatican
Fontana di Trevi
Fontana dei Quattro Fiumi at Piazza Navona
Santa Maria sopra Minerva Rick Steves Rome 2007 I like his books a lot. Very down to earth and cuts through the typical tourist bs. Actually met him in London at a restaurant that he had recommended and he was back to check on it.
My first questions would be, where are you in Esquilino and why did you choose that area? It all depends how closer/far you are to Termini and in which direction. There are parts of that area that I wouldn't touch at all. Did you have other options? A lot of tourists like to stay near Termini because of the abundance of hotels and the lower prices but it's just not the best neighborhood.
That being said, there's tons to do. What do you like and maybe I can point you in the right direction. I have family in Rome and visit often so I can help a bit. Also, a couple of weeks ago, I posted an entry on Rome in my blog:
Be sure to go to the Vatican to see where the church spends all the money it grifts from the poor. Most will never get to see the gold mosaics and lavish marble, but what the hell, at least they've paid for their ticket to heaven, right? Pope needs a new pair of Gucci shoes.
Also Spanish Steps, Piazza Navona, Piazza de Popolo and walk up Paletine Hill, it has an allsome view overlooking the Forum.
Husband and I have been four times... took inlaws a few years ago. All above are good suggestions. The best hour we've ever spent there was the SCAVI TOUR. It takes you underneath San Pietro to the necropolis the basicalla was built on. They only take about 10 down at a time and reservations are booked well in advance. If you are interested, I'd book now. It only costs 10 bucks and you pay when you get there.
I've heard reservations can also be tricky to get. It seems you are at the whims of whomever gets the email. We were fortunate and knew a cousin of a mosignor that worked there... he hooked us up.
My inlaws said this was their favorite thing they did in Europe on the 28 day trip (and they are not Catholic!).
one more thing... when you go visit san pietro, get there at 7am - or whatever time it opens that time of year.... there will be no tourists nor crowds. It is silent and virtually empty except for a few praying nuns/priests.
it is magnificent and awe inspiring, at least for us.