What is your fun, favorite and cheap way to entertain yourself and your family? I like to go swimming. I live in a lake community, and in the summertime, being able to swim is one of my favorite things to do. We pay for our dock fees and community manintenance once per year, the rest of the time it is free. So, it is a great thing to take advantage of especially if you like to swim.
The cheapest way to entertain yourself and your family is to take the kids to a library from time to time. Since fewer people go the libraries these days, it's something the kids will enjoy immensely. If you don't want to leave the house then you could try solving jigsaw puzzles or play board games.
I think availing of the free wifi in public areas is a free way to get yourself entertained. I also think watching one of those free celebrity mall tours im the mall if you have them is a free way to get entertained too.
Other than book and movies, arts and crafts. Not just drawing and painting but the whole new world of mixed media and assemblage. Old packed become pop art robots, old magazines collages, broken crockery new mosaic covered paving stones. It is amazing how all your old junk can become new and much more fun... well, probably junk. But I never did have an overly refined interior design aesthetic. And it is a lot of fun.
Public art galleries. Especially these days, when digital forms of art are becoming increasingly more prominent - there are some really cool works which may grab the attention of kids as well, which incorporate colored lights, projections, installation art, holographs and other neat stuff. In most states throughout the country these are free and or offer free sections. It's usually only the special exhibitions that they charge an entrance fee for. Photography - take a trip to some nearby park and spend a day taking pictures of the plants, flowers, trees, bugs and animals, while you get some fresh air and exercise. Bowling - it's dirt cheap and can provide hours of entertainment, especially if you go during their off hours when they aren't busy.
When it comes to entertainment that involves the entirety of my immediate family, there really is not a cheap option. There are too many of us. There are six of us living in this house, three of which are young children. So if we ever want to do things as a family it is never cheap. We often just fill the car up with gas, and drive to the nearest seaside. Of course you buy things whilst you are there like toys and food.
My favorite is potlucks. And then whoever's house it is will be responsible for for figuring out what we'll do afterwards. Its a lot cheaper to if you only have to prepare one dish, and get an entire meal out of it.
For the most part I just have to worry about entertaining myself, so it’s pretty easy for me. Libraries, parks, super-inexpensive classes at the Y or similar places, the very occasional beach trip… I would probably opt for similar things if I had young children, though. Getting creative, like @Penny said, as well. I’ve been getting more into that just for kicks, and it’s a great way to use things up that’d otherwise just end up being thrown out.