Gunpowderplot Dartmoor is great for free camping BUT there are rules some a lot stricter than the usual campsites www.dartmoor.gov.uk/visiting/vi-planningyourvisit/camping
As for other activities you can do for free try this list
national trust did 30 days wild www.mywildlife.org.uk/30dayswild/ you will obviously have to arrange this yourself. But a list of a few things to do and a park you are away.
Litter pick! The local conservation group here arranges big litter picks in our woodlands. Try that with the kids they are kid friendly and provide all gear you need including pickers and gloves. Its not glamorous but its the feel good factor of caring for the environment. The group also runs bug hunts with an expert, guided walks, den building afternoons. All these are free!
Our museum runs free craft days during half terms, guided tours of the collections and story telling at xmas. Sign up to your local museum email list or follow the twitter account.
In fact I'd suggest getting a twitter account and follow as many of the local attractions/museums/librarys and galleries as you can to find free events. More often than not they retweet other local organisations increasing your pool of activities.
The library will also contain a ton of ideas for free days out. Their mailing lists/twitter feed will keep you up to date with all thats going on.
You local council events page. Keep an eye on that or again sign up to the mailing list. My council sends out a weekly email of everything that is going on in the area. I can pick and choose.
Don't dismiss things like carnivals, Pride events or any kind of 'festival' your area has. Usually there are many things to do.
Here we have pride, food, music, history and arts festivals through the year with free and paid for events along the way.
i'm also lucky here to have a playscheme thats free for kids. you just have to turn up!
Also have a look for a scrap store near you www.scrapstoresuk.org/ they too can run free workshops for kids and adults. Its also a great way of getting materials cheap or even hiring art kits for a day in doing something new. Much much cheaper than if you were to buy it all yourself.
I hired out a batik kit for a week and it cost me £15 plus the wax. Thats way below what it would cost to buy the items new.