The new forest is an obvious one. Lovely walks and it’s always novel to see the ponies, pigs and cows roaming free.
beaulieu Motor museum is quite interesting if you like that sort of thing.
If you have national trust membership then mottisfont is a nice place to visit.
Moors valley country park is also lovely, they have go ape there too depending on your DC ages, but you don’t have to do it, still plenty for younger kids.
There’s lots of nice beaches along the south coast and if you get a rainy day, splashdown in Poole (I know it’s Dorset, but not too far) is a good indoor water park.
You can also get the boat from Poole harbour to brown sea island, again national trust, but that’s a lovely day out.
It’s a lovely part of the world. Have fun.