Our local library has free passes to many different activities like museums, state parks and beaches etc. and all you have to do is sign up for the passes when you need them. Also a lot of museums will offer discounts on certain days. We got to go to the Boston Aquarium at a huge discount if we brought in an empty pepsi can (it was some kind of promotion) so I am sure other places in other states do the same thing. Sometimes buying a season pass to a place you go frequently can save lots of money and newspapers often have coupons to different activities and events.
Other things you can do free are things like in winter, build snowmen or snow forts or go snow shoeing or for walks or go to the local library for story time (most libraries do this).
In summer there are picnics, free fireworks displays, fishing, swimming (or sprinklers), walks, bike rides, hiking, playgrounds and many towns offer free concerts in the park (sometimes the music is not everybody's taste but it is fun to get out and people watch and you can bring some snacks and drinks and just expose the kids to different kinds of music etc.) Adult education often offers very cheap workshops in all kinds of things such as using a computer, needlepoint, fly tying, cooking, woodworking, sewing etc.