One of the only parts of the great city of Rome that has retained its cobbled roadways, Trastevere is a haven for expatriates, where you'll hear an alluring mixture of languages flowing from umbrella'd tables as you wander these narrow streets. Located across the Tiber River from central Rome, Trastevere is an Italian world unto itself (literally meaning “across the river”), off the beaten path from the iconic structures of Rome and also away from where many of the tourists will be. There is great food and nightlife, as well as a handful of landmarks (Villa Pamphili, Janiculum Hill, and the ancient Santa Maria on the piazza of the same name).
Some of the best Italian eateries in the city can be found in Trastevere, and while some great places are concentrated near Via Benedetto (delicious pizza and Fiori fritti at Dar Poeta and Bir & Fud, authentic pasta at Taverna Trilussa), an hour of spontaneous wandering out from the Piazza San Callisto will run you aground of excellent dining.
As for places to grab a glass of house wine or a well-mixed cocktail, there are a high concentration of spots at both ends of Vicolo del Cinque, while the aforementioned Via Benedetto also is home to great pubs like Mate Bar and Ma Che Siete Venuti a Fa. Again, Trastevere is a district for wanderers, so take your time; even if you have to take that four hundred year old church out of your itinerary. It'll be worth it.