What type of site do you want to see? but generally speaking you will never be far from a country House if thats what you are looking for.
Beaulieu is part of the new national park the new forest, gets very busy in the summer. its also close to Southampton and Portsmouth, both of which have navel history and related sites. HMS Victory, part of the Nelsons fleet is worth a look.
Goodwood, has no measum, but is very much worth a visit. Goodwood house is close by and is open to the public. The market towns of Midhurst & Petworth are quaint. there is also Petworth country house this is open to the public.
Portsmouth and Brighton are also within easy reach.
The track is also a working airfield, so access is always possible. there is a small memorial garden there as well.
Brooklands has a good measum, this can somtimes be overun by kids but is well laid out. Sadly not much of the track is now visable. Places like Kew Gardens & Richmond Park are close as well.