Congratulations on your first trip! I'm sure you will love it!
Like the previous poster said hostels are the cheapest etc.
You can fly cheap if you get flights ahead of time, or you can do overnight trains if you want to, but they may actually be more expensive then the cheapest flighst you can find.
Once you are within Italy I would definitly recommend trains to move between the cities.
Another good tip for going on a budget is to eat from supermarkets at least once a day. On my most recent trip, for any city I was in for more then a day, I hit up a local superamrket and bought breakfast foods and waters as well as some snack food that I put in my day pack so I wasn't tempted to buy a 5Euro water from a snackstand outside the Vatican or something. This allowed me to splurge on a few nicer dinners as well, and some days when I was really busy sightseeing I would just snack from my pack on the subway between sites and skip lunch altogether.
Hope that helps!