We will be staying in an apartment in Trastevere at the end of June - what is "not to be misseed" there?
Pizza and Gelato. Really - we traipsed over to Trastevere each night after pizza and gelato. Also did head up the Gianicolo Hill, saw the neat chapel up there and the memorials of the unification martyrs - most interesting - and the nice easy entrance to Vatican City.
St Maria in Trastevere - one of Rome's oldest churches with beautiful mosaics. Villa Farensina - a Renaissance villa with decorations by Raphael. Climb Gianicolo Hill for a view over Rome Take Rick Steve's Jewish Ghetto Walk. You will get a look at Ancient Rome with the ruins of a temples. Note Teatro Marcellois a Roman theatre, with "newer" apartments (from the 15th century) built along the top tier. Lots of restaurants in Trastevere. I found a small cafe that served the best porchetta panino (roast pork sandwich) - it was so large it did me for lunch and supper and cost only a few Euro. The meat is cut off a whole roasted pig - yum! It was in a side street off Lungara.
Dar Poeta is a good pizzeria there, but I have heard La Gatta Mangione is even better. (Reservations required. We were turned away on a Saturday night in December!) Cacio e pepe at Roma Sparita is to die for! Gianicolo (Janiculum) walk is excellent. There's a good self-guided walk in the book "24 Great Walks in Rome." Pick up a copy and be delighted! It is easy to get from Trastevere to the rest of the Centro Storico. We especially like the tram #8 that takes you to Large di Torre Argentina. The church mentioned by Leslie is lovely, too. Rick Steves' book has a Trastevere walk in it, but it's not as comprehenive as the book I mentioned above. The Ghetto walk in RS book is worthwhile, though.
My husband and I stayed in this neighborhood for nine days in December 2011 and we loved it!!! Don't miss Saint Cecilia's. And I second the tip on Villa Farnesina. Dar Poeta is fantastico! I speak some Italian but they're so used to American students they started speaking English as soon as they walked up to our table and heard us talking. And if you're a night person, check out Big Mama on Vicolo C. Francesco a Ripa 18. It's a blues club in Rome and we had a wonderful evening. It's easy to get to with a map off the number 8 tram. Have a wonderful time!
I echo the St. Cecilia, the Santa Maria in Trastevere, including that piazza, and Villa Farnesina is incredible, with Raffaelo's frescoes-not revolving around christianity. Lots of great food, including a great pizza place--I wish I could remember what street it's on--a two way large thoroughfare on a bus line that crosses the Tiber. Open late, cafeteria style.
I've heard that Campo Firo (sp?) is a hot spot: farmer's market by day, music and night-life by night. It's in the Trastevere neighborhood close to my apartment; I'll be studying in Rome from June 24 - Aug 3.