Deborah,
We just had a weekend at the Caravan Club "CC" Bladon Chains site just North of Oxford, which may be a bit close but at least you could pop home if you had forgotten something.
Bladon Chains has a mix of grass and hard-standing pitches, and what made our arrival special at an almost full site was one of the wardens placing a levelling block under one of our wheels as we drove onto the pitch. The guy didn't need to do that, but I just thought nice!
But I guess you want to go a bit further. Why not try the CC Sheepcote Valley site at Brighton? If you avoid the rush hour, it's only 2 or 3 hours from Oxford to Brighton, and also a great site. And loads to do in Brighton - about 1/2 hour walk into the centre.
Or the really nice CC Carsington Water site near Matlock Bath?
A recurring theme is the CC - all of their facilities are uniformly good, the wardens are really helpful and friendly. Later on you may get a bit bored with the sameness of the toilet blocks, then take a look at the "Best of British" sites.
Robert