So jealous! Rome is amazing! I did a lot of research on kosher options when I went two years ago, so some things may have changed. I found that most kosher eateries and stores were concentrated in the Jewish ghetto area, and another Jewish neighborhood, Piazza Bologna, that is quite far from the center of town. If you are mostly going to be in the center of Rome doing touristy things, especially if you don't have a car, it will save you time to stay and eat in the Jewish ghetto. (If you have a car, there is a kosher grocery store in Trastevere with amazing products like braesola and delicious kosher cheeses, but it is difficult to get there via public transportation.)
The Jewish ghetto is the neighborhood around the synagogue/Jewish museum. On it's main street, there are a variety of meat and dairy restaurants as well as a grocery store and a few other stores. One place I remember being really good was Ba'ghetto Milk (I had a pizza with squash blossoms on it that was insanely good. Also make sure to try Artichokes ala Judea, the "Jewish-style" artichoke dish.) There is a bakery called il Mondo de Laura that makes beautiful cookies that are great for snacks or souvenirs. There was another restaurant, towards the end of the main street, that had a typical Italian spread of appetizers and snacks that come free when you buy a drinks special, it's an Italian happy-hour style tradition. I didn't try it, but it looked like fun.
Wine is relatively cheap at nearly every store, so you can buy a bottle of wine and some cheese and bread or snacks and have a picnic at Piazza Navona or some other beautiful place while sipping wine. It feels very Italian :)