I'm going in February to watch a champions league game and I've been there 4 times as I have a cousin who lives in the center.
I could tell you that if you decide to visit the tourist destinations, you should definitely see the Sistine Chapel as it was my favorite place. The subway is very efficient to get around.
Also if you go eat, I'd avoid tourist spots. I could ask my cousin for good restaurants that locals go to as the tourist spots are not that good. The Jewish ghetto is very nice there.