We travel a ton and I just posted about something else travel related which is funny. I'm probably going to reiterate what others said, but here is what we do for Europe and some general tips:
Some airports are super expensive to fly into or out of. Like you can one way rental a car on each end and save the money.
They hose you on checked bags. One of our credit cards is an airline card for free checked baggage.
Messing with days can save a lot. Last night I found out the superbloom is probably a go in death valley this year. I bought tickets. In the week we had open they ranged from $400 to $2500 for two. That's with the same airline we had vouchers for.
For Europe:
It's always expensive in the summer. Go to google flights. Don't put a destination in. Then do a search. If you can play with dates it gets even better. You don't even need to do a date. Half our trips are "Where can we fly stupid cheap in the next 4 months" You don't need to be in Italy. You need to be in Europe. Once you're there travel is generally cheap. You can get stupid cheap tickets doing this. On a whim last year we went to London/Bristol/Bath for a week and a half. I randomly looked and I found tickets direct to Heathrow for $395 each all in. I told a friend about it and they took a 3 day weekend to London. Look up how much train tickets are too. You see some cool stuff.
You mentioned Italy. It's been a while but the last time I went there I found a ticket to Edinburgh for about $700. Then I took a train down to England and got a flight out of Durham to Rome for like $150. Durham happened to have stupid cheap flights at that time. I do better with this manual looking than relying entirely on search websites. Flights from where I flew out of to rome were about 2x as much as I spent doing my research, adding an overnight in Durham (cool castle, neat cathedral), and being on a train a couple hours.
When we fly to Europe we never take a checked bag. Never ever. If we want to bring must check stuff back we get a cheap suitcase there. We're to the point where all we take is the largest sized backpack that goes under the seat.
Oh yeah. If you're not up for an adventure, don't fly in Europe on some of the super budget airlines. You can google who they are. I've got some hilarious stories. My favorite is a flight some tiny airport in eastern europe that was seat everyone at once. They opened the doors to the tarmac and everyone took off running 100 yards to the plane holding their stuff. It was great. They have their own charm.