Go to Google Maps, zoom into the area near your hotel, and in the search box enter LAVANDERIA or also LAVANDERIA AUTOMATICA.
The Lavanderia Automatica is a self service wash and dry, others you can leave your stuff and pick it up later. I think they charge about 7 or 8€ per kg (1kg=2.2 lbs). If you go to a self service, you might pay 5€ for a standard wash, and another 5€ for the dryer. The equipment is large, like in the US, therefore one standard wash might be enough for all you have.
The hotels might charge by the piece, and at 4 to 10€ a piece, it adds up. However, as said above, doing laundry on the go is way more convenient than carrying excessive luggage. And if you use trains, you must really strive to travel with a carry on suitcase and a backpack only, because train travel and heavy luggage don’t go well together. On trains you don’t check in luggage, you have to schlepp it on board by yourself and place it on the overhead compartment. There are also luggage holds on each side of every car for larger suitcases, but they fill quickly. Also those luggage holds are not necessarily visible from your seat, therefore you should buy cable locks like these to secure them (tie them together) to prevent someone taking your luggage at some intermediate stop, while you are not looking.
Below some prices for a lavanderia (not a self service laundry mat) in Florence near the Central Market
http://www.lavanderiamanfredi.it/listino.html