Rent: Like Nardini said, it's going to depend. It will be similar to the US though. Big premium spots = a lot, while not so great spots = a little. The good thing about Italy is, good spots are relatively close and it's usually beautiful no matter your surroundings
Electricity: More than the US for sure
Gas House: Deffinetly more than US
Gas Car: A lot more than US
Car Insurance: - don't know -
Hair cuts: I would suggest learning how to cut each others. Will save money no matter where you are.
Income Taxes (I'm confused about how we'd get taxed): - don't know -
Health Care: - don't know -
Average grocery bill (mostly eating in): Learn to like Italian cousine and it'll be a lot cheaper.

Europe is just all around more expensive. The US is rich in resources and land so it's fairly cheap compared to other nations. When moving abroad you should almost always expect some costs to go up.