I would only take a token amount of cash (20-30 euro for small purchases that might not accept cards) and plan to spend on my card (just make sure you have one that doesn't charge fees and preferably no exchange rate loading).
I agree that Rome is as cheap or expensive as you make it and even in the obvious tourist traps, it was fine - main thing is watch out for pick pockets and avoid street hawkers. A man started talking to me near the Coliseum and I'm not sure what his game was but I'm sure he was trying to trick me into buying something off him, not sure what, he didn't have any 'goods' that I could see.
But I bought an ice cream from the Kiosk next to the Trevi Fountain, where there was no prices on display and I was braced for it being expensive, but it was 3 euro for a small one, so not the cheapest ice cream I've ever had but not a stupid price either.
Restaurant prices seemed similar to UK mid range independents unless you deliberately seek out a high end place, wine possibly slightly cheaper.
You can see loads by just wandering around centrally and just stopping for lunch, dinner, a drink etc as needed. I think you have to book ages in advance to get inside the Coliseum and it will probably take quite a bit of your time, so you might decide it's not worth it on a short trip but the area outside is worth walking around, although that's where all the con artists hang around.
Which airport are you flying into? If it's Ciampino, which unusually for Ryanair is actually closer than the main airport, is tiny so you don't need to allow anywhere near the normal 2 hours before departure, which means you can stay in the city longer.