If it's not liquid you can get it on the plane in the US, but Italy has requirements around meat and dairy products. Bars, pasta, etc should be fine. I also see no restrictions on produce, so fruits and veggies are also not a problem. The meat or cheese based items would be fine to eat on the plane, but dispose of before arrival.
I wouldn't worry about bringing a ton. Enough to eat on the plane in case there is a problem, and some snacks for if you are delayed. I travel GF now and my son has been for a long time. It's annoying, but at any airport of any size there are shops that sell things that are GF - salads, yogurts, fruits. Once you are in your destination, grocery stores should have plenty of naturally GF foods, so you won't starve. Finding specific GF foods and restaurants will take some research but is not a problem in most places.
Again, your biggest concern will be while in transit and immediately after arrival, so bring food for that but don't worry too much about having your own food for the rest of the time - you can find GF food all over Italy. I do always rent apartments or aparthotels with kitchens, just in case so I can cook and/or store things like cheese and meats. It's a big help but not at all necessary.
Learn the local label for GF foods, and note that they may be sold in pharmacies because celiac is a medical condition. Also google Gluten Free [destination]. A search for Gluten Free Venice brings up a specific GF store, some restaurants, and a lot of other resources. I'm using Venice because I'm going next year, but the same should apply to most cities in Italy.