I should clarify that there are self service laundry places that will cost less but you have to weigh that against what it costs you in your time.
For example, for us it was worth the 20 euro to drop the laundry off at the service and then spend the time we would have spent waiting at a laundry mat and take the 2 to 2.5 hours and head to the Baths of Caracalla and explore the ancient ruins. For us exploring an ancient Roman bath house was worth the extra money but for you it might not be. That is something you will have to figure out.
Travel was done exclusively by train, bus, taxi and our legs. The Trenitalia app is something you should DL before hand and familiarize yourself with ahead of time. It is an invaluable tool when traveling from city to city. Don't discount taxis either. While more expensive than public transit they can be a quick and somewhat cheap way to get to a place fast especially in a larger city like Rome.
Our legs did most of the heavy lifting on this trip.
@DigDog nailed it again when it comes to the pick pocketing. IMO it is WAY overblown as long as you play it smart. I use a Ridge wallet and kept it in my front left pocket at all times. The only items in my wallet were my US drivers license, my two credit cards, and a couple hundred Euro. Use tap to pay via your phone if your phone supports it and DON'T carry your passport on you.
We felt completely safe during our time in Italy. At no point during our visit did we feel otherwise whether we were in a big city like Rome and Florence or a smaller city like Monterosso al Mare. 100% of our interactions with the people of Italy were positive. Everyone was very friendly.