The New Forest, if that's not too far. Butser Ancient Farm (off the A3). The Weald and Downland Open Air Museum (near Goodwood), and the gardens at West Dean which are nearby. Nymans (National Trust, wonderful gardens) and Leonardslee garden, both near Horsham. Wakehurst Place, the 'out of town' bit of the Royal Botanic Gardens, near Crawley. These are all to the South of Camberley, as I know that area better.
Around Camberley and Aldershot itself are several good military museums of various types and the flying stuff at Farnborough. To the North you have Windsor and Eton, and Runnymede I rather like too.
There are lots more places like these, what sort of places are you interested in?
If you really want to go shopping, you could always drive through the New Forest to Poole and visit Robert White...
Enjoy the trip,
Peter