Been there several times. Had an amazing time. Once been for two weeks, which gives enough time to go off track a bit and enjoy other things than the main tourist attractions.
I'd especially recommend two places: (See map below), one is an icecream shop and the other is an italian restaurant (I think it's called Ristorante La Rustichella ). They are both very popular with locals. It's worth to go in the evening, the piazza in front is very lively around that time with locals. Been there 5 years ago or somthing so I really hope both still exist.
The easiest way to go there is to go to the Vatican Museum, then (when facing the main entrance) walk the street to the right untill the second pair of stairs across the street. Go down there and you'll come on to a little piazza with a small church. Directly on the lest corner is the ice cream shop and about 3-5houses further is the Italian restaurant.
If you don't want Italian foor for a night you can also walk that street a bit more, pass a flower stall and got to the right, there is an amazing Chinese restaurant.
Somewhere around there is also a big market, I think it's towards that big road (Via Cipro) and than turn right at the fountain/ roundabout. Lovely fresh products (Take some fruit, balsamico, bread and ricotta cheese, take a cold white wine/beer and I assure you, you want your vacation to last another few months!
). A lot of locals come here to do their shopping. And if you follow that road you get to another piazza with some small shops and a market. Very authentic and you'd hardly find a tourist. Just remember that that also means hardly anyone speaks english; just use gestures and know a few words Italian and you'd be more than fine.
The town of Ostia is also very nice. Not too full (at least the times I've been) and some nice cafés at the boulevard. It might be nice to combine that with a trip to Ostia Antica, well the remains of it. Do watch out here, on a warm day it can feel here like 40+ degrees easily, so drink enough!
You can go there by train which is really not a nice way to go there (think Indian trains!) with the heat and packed with locals. But it's worth it.
In Rome, have a few hours at Villa Borghese (The park, not the house itself). Especially on warm days it's very nice there with a lot of shade, fields for picknicks and water/fountains.
SEE the second map. at the arrow is one of the foutains. You can get there by bus.
Can't think of any other tourist-free places at the moment, but if something comes up, I'll post.
Have fun! (I'm jealous!)
