We see this every day in this forum: Everyone wants to visit here, and is looking for “cheap/affordable and safe". However, such a place just doesn’t exist; the two terms are mutually exclusive. It's all about supply and demand: If it's in SoCal, then it's not going to be cheap to begin with. If it's a safe area, then it's more expensive. Average price is going to be in the $150 range in summer. $55? This ain't Calgary, buckaroo. It's LA. $55 means you sleep with Templeton the Rat and his cockroach and bedbug friends. And hope the hookers in the next room don't keep you awake all night. Sorry, that's just the financial reality here in LA.

LA is probably the only city where you DON'T want to stay downtown.
Since LA is so spread out, there is no single place that is close to EVERYTHING. Most people who come here tend to stay in Hollywood, Santa Monica, or Anaheim. In Hollywood, I recommend the Orchid Suites, Hollywood Celebrity Hotel, or Holiday Inn Walk of Fame, all near Hollywood and Highland. Another one is the Best Western Sunset Plaza. In Santa Monica, the Holiday Inn Santa Monica is close to the pier and reasonably priced. In Anaheim, I recommend the Howard Johnson (HoJo) Anaheim, and the Candy Cane Inn. There is also a nice place in Burbank called the Best Western Media Center, very close to Warner, NBC, and Universal Studios.

In addition to the whole pension/hostel situation you could look into apartments. They are generally clean and can be centrally located. Even if you only cook breakfast, they can be perfect for that, and larger and more comfortable than many rooms. Just search the city in question, and there should be all sorts of options for lodging.

I would probably use both cities as a home-base and explore some of the smaller towns outside by train. I think renting bikes to do the Tuscan countryside is an excellent idea! I don't know how hard you can push the smaller child, though.

I would definitely try to spend a week in each city, using Florence as a home base for exploring Tuscany. You don't really need more than two days in Florence to see the basics, but a full week for Rome, for sure. (Oh, and you don't want to be driving in Rome at ALL!!! They have a reputation as the world's worst drivers for a reason…red lights, etc. are considered suggestions, and the last thing you want is to get stuck in one of those huge roundabouts, without knowing where to get off. If you DO decide to rent, DON'T take the car into the city, garage it outside of the central part of town, near both a bus stop or metro station and a highway, that way you avoid driving in the city.)

South Station is right on top of the Red and Silver Lines, and only a few stops from the other lines so you can get most places in Boston fairly quickly. Most of the historic areas are linked by the Freedom Trail. For small kids, the Children's Museum is quite near South Station. There's also a walking tour of the Freedom Trail geared to kids, but you can get a map and do it yourself for free. The Aquarium and Science Museums are wonderful, and both are on the T. The Harvard Museums have a couple of good ones for kids. The USS Constitution is wonderful.

Food on a budget — Quincy Market/Faneuil Hall is a massive and unusual food court. Chinatown has many good inexpensive places. Generally the ethnic places (Indian, Thai, Chinese, Mexican) have good cheap food if your kids will eat it.

Also the cheapest harbor cruise is the Water Shuttle from the Aquarium/Long Wharf to the Charlestown Navy Yard (where the USS Constitution is). Fabulous views, $2.00 each or free with a subway pass.

If you live in a MA town (and maybe CT or RI), check your local library to see if they have passes to any museums.

Everybody (but me) loves the Duck boats.

I think there's a Comfort Inn near Alexandria…near the Van Dorn Street Metro station (about half-mile from the metro). Go online to one of the travel sites and look. Be sure to book your room early to get the best price! If you wait till late, the prices go up (just like airfares).

They have a shuttle van that will take you to the Metro an also pick you up when you call them to return to the hotel.

Right now the rates are about $140 a night for a group of up to 4 people.