The daily market held in Campo dei Fiori itself is a given. Stroll, look, watch, buy.
SantAndrea della Valle is at the corner of Via dei Chiavari and Corso Emanuele II, and it has the highest dome in Rome after St. Peters. The interior is magnificent, with absolutely huge frescoes featuring scenes from the Life of St. Andrew; the interior of the dome is amazing too. I spent a nice chunk of time wandering through it; the audio guide at the front gives a good overview of all its treasures, which are numbered.
Have a great cappuccino in the morning or an evening apertivo at Bar Farnese on Via dei Baullari. They are incredibly sweet there and their coffee is absolutely excellent. Sitting there you can enjoy the incredible view of Palazzo Farnese, which is the residence of the French Embassy, known to be the most magnificent Renaissance palazzo in Rome. It is now possible to visit Monday and Thursday. The 4 p.m. tour is in French; the 5 p.m. tour is in Italian. To make reservations send email to firstname.lastname@example.org Just make sure you bring all the documentation with you for entrance into the Embassy. There are, however, two lovely matching fountains in the square in front. I enjoy sitting there and watching people come and go.
I think one of the best things to see in the area is the the Galleria Spada, inside the Palazzo Spada; it is in Piazza Capo di Ferro off of Vicolo deVenti. The Palazzo was designed by Borromini; the lower rooms are closed to the public, but the upper floor contains four rooms with the private family collection of paintings and sculpture. You can get a good idea of what is there to see on this page. Massimo told me to visit there, and I am glad I did...it was a quiet, peaceful haven in the midst of a very busy neighborhood.
Be sure to visit Antico Forno Marco Roscioli on Via dei Chiavari. If you dont buy a loaf of bread, get a piece of pizza bianca or pizza marinara to go, it is heavenly. Where Via dei Chiavari meets Via di Guibbonari, the family also operates a beautiful enoteca/salumeria. In the front you can buy wines from every region of Italy, plus beautiful cured meats and cheeses, and in the back, the enoteca serves absolutely excellent food.
Grappolo DOro Zampano is in Piazza della Cancelleria and they serve a good pizza. Nearby is the very popular trattoria Ditirambo, which is lovely inside, but the service can be rushed; the menu can be a solid hit or a near miss (which I found somewhat tragic). If you are willing to go a bit off the beaten track, there is a terrific restaurant at the end of a side street off the Campo on Vicolo delle Grotte, Da Sergio.
I went to a delightful enoteca in the Ghetto, in Piazza Margana, called Vinando. It was quite wonderful...great food, from pastas and secondi to different antipasti and primi, salumi and cheeses. It is open late, has a very modern vibe to it, and after midnight, one of the owners plays piano and sings!
My other recommendation is to take a tour with Context:Rome: I take one every time I visit Rome.
A last address: the nearest supermarket, Punto SMA, is on the west side of Via del Monte della Farina, between Vicolo dei Chiodaroli and Via del Sudario.
Those are my Campo dei Fiori best bets!