Hi Jess,
Here are some suggestions, all of which I considered to be very safe:
Venice - Alloggi Barbaria - My friend and I stayed here in June 2013 and found it really comfortable. We paid 110 euro per night for a double. No canal views but a wonderful family who is very welcoming and helpful. Listed in Italy
Cinque Terre -
Vernazza - Francamaria Rooms - We stayed here in June 2010. I can't remember the rates. Some steps but not too many. I think they reopened following the floods.
Monterosso - Il Timone Loreto - We stayed here in June 2013, also. Located in the "old town" near the post office. We paid 90 euro per night for a double. The family owns several rooms all on the same floor of an apartment building, so you would probably have no trouble finding room for 3.
Manarola - This summer I will stay in the Hostel in Manarola - no reviews yet! :)
Florence - Hostel Archi Rossi - We stayed in a private room in the "Annex". There were lots of steps but well worth it! This was in 2010, too, so I do not remember the rates.
Rome - The Beehive - In June 2010, we stayed in a dorm, but they also have private rooms. This place books very quickly, so make reservations early. This is a Cross-Pollinate property mentioned by Claire in a previous post.
Montepulciano - Palazzo Bellarmino - We rented an apartment here in June 2010. It was plenty big for 3.
Have a great trip!
Sharon