Though often referred to as to as 'couchsurfing' (because of the famous site of the same name), many who offer a 'couch' for the night actually have a bed for you to sleep in. They rarely charge anything and are also very keen to show you around the city. What more could you ask for?
Even if you don't need a place to sleep, couchsurfers often like to act as tour guides, showing you around the city you're visiting.
Note that many couchsurfers don't want to be used as 'free accommodation'. Couchsurfing is a social thing - guests that leave early in the morning and come back late at night and don't interact with their hosts aren't appreciated. There are two main couchsurfing websites:
Home Exchange - Couchsurfing for Grown-Ups!
If couchsurfing sounds too 'studenty' for you, try a home exchange instead. For a small fee (about the price of one night's accommodation in a medium-priced hotel) the Home Exchange website allows you to communicate with people in your chosen country and arrange a short-term exchange - they come and stay in your home, you stay in theirs. Usually this is done at the same time, but if both houses have spare rooms, you could travel at a different times and host your new-found friends in person. What's more, after you've paid the registration fee, you can use the service as often as you like.
Visit: Home Exchange