Keep in mind that the water from the fountains is good to drink. Bring your own Nalgene or other reusable water bottle from home and fill freely/and free!
As far as snacks go, hit a grocery store and buy what sounds good. I usually go to Italy with my own creature comfort snacks, dried fruit, nuts and Lara Bars, and a bit of my favorite dark chocolate. Snacks are something I find to cost less to bring from home, and I create space in my carry on going home as I've eaten my own food. (Not to say that I don't shop the cookie aisle at the Co-op for great Italian cookies!)
You will not be able to eat at important sights... i.e., don't get caught eating on the Spanish Steps or in the Forum...you can get a ticket!