I lived in the Rome area for three years and there is tons of good advice here already.
A piece of general advice I would give is to not dine near the popular tourist attractions. The food will often be mediocre. At dinner time don't find yourself looking for a restaurant in Piazza Navona or a place like that. Wander over to Trastevere (as many have suggested). You will get a much more genuine experience.
Some specifics on gelato, there are flavors that are common there that we don't get in our iced cream here. My favorites were nocciolo (hazlenut) and amarena (sour cherry).
Choose a day to see the Sistine Chapel and go there first thing. There are no reservations and the line can get crazy long.
Finally, and I don't want to be a Debbie Downer, but you must be aware of your surroundings. Pick-pocketing is rampant in Rome and it can ruin your trip. They have developed it into an art form over there and even when you think you are being careful they can get you. My mother was clutching her purse under her arm going up the escalator from the subway and someone behind her adroitly cut a slit in the back of her purse, reached in and stole her wallet (I had told her NOT to carry her purse around). It all happened within the 15 second escalator ride and she never felt a thing. When you go out for the day, bring as little as humanly possible. And no ID! You won't need it. Bring a little cash and one credit card and leave the rest locked in your hotel room safe. What you do bring with you is best kept in a zipper pouch hanging from your neck and under your shirt. Even this isn't foolproof but its about the best you can do. Beware of anyone approaching you to "help" with a map or some other form of distraction. Or groups of young kids that gather around you; they will jostle you so you can't feel what they are doing. Wave them away immediately. I would occasionally have to go to the embassy for residency paperwork or have pages added to my passport and there was always a waiting room full of forlorn American tourists who had had their license, passport and money stolen. A very sad scene. I don't want to be unnecessarily alarmist about this but I've known too many people that thought it wouldn't happen to them and then it did. And if you are taking an over night "robber train" (this is their term for them, not mine) from one place to the next, well, I would try and avoid doing that.
Back to the good stuff and there is plenty of it. Of the places you are going, Rome would easily be my favorite followed by Venice. There is so much history to see in Rome and Venice is so unique and just has a magical feel to it. Florence is great if you're an art lover/appreciator. You can see more masterpieces here than the rest of the world combined.
If you have the means while in Rome, I would suggest taking a day to go a little south to Nemi and Castel Gandolfo (summer palace for the Pope). These are smallish, charming towns overlooking lakes. They will be less crowded and touristy but pretty neat. Have an authentic lunch perched high above one of the lakes.
