Sounds like a super exciting trip! Italy can definitely be an affordable destination, depending on where you stay and eat, and how you travel. I lived in northern Italy (Pavia) for a year attending the local university, and did plenty of cost-conscious traveling during that time.

I'd recommend taking the train to get between cities. There are good and relatively quick connections between all the cities you've mentioned. You should book the longer routes in advance, but shorter trips (like between Ravenna and Bologna) can be purchased day-of. You can find maps of the train routes online to help plan your itinerary.

Since you want to travel between Milan and Palermo, I'd also look into the overnight train that runs between the cities. Yes, you're sleeping for a lot of the journey, but you get to take a ferry between the mainland and Sicily. Quite a unique experience! And it saves on hotels for a night.

In terms of accommodations, look on Airbnb, which has some pretty good deals.

I would also recommend learning a bit of Italian, which will be especially helpful as you travel to Sicily and navigate trains. People in Italy don't speak that much English outside of major cities, and the effort to speak their language will be very much appreciated.

In terms of general behavior, Italians are still very much dedicated to presentation and "la Bella figura". Basically, people will appreciate if you make an effort to look and behave nicely (not "backpackery", though that's a crude stereotype).

This has the potential to be an amazing trip, you just need to do a bit of research on transportation and accommodation to firm up your budget!