Square & compass pub in worth matravers, studland beach, corfe castle, monkey world, Durdle door / man o'war. (Best bits of Dorset).
Most places along the coast of Devon/Cornwall (skip the bigger towns/cities).
Brighton if you like party towns with a bohemian feel.
Skip everywhere between Portsmouth and Southampton (there are a couple of nice places, but it's all very industrial/military around there)
New forest is pretty in parts, avoid on the weekends - it gets very busy! It actually has more horses and cows than actual trees
Stonehenge is smaller than it looks in the pictures (it's still cool to see, not sure it would be high on my list to stop at, but if all else fails you can drive past it!)
Winchester is a beautiful town if you need somewhere to stay the night. (I'd say the same about oxford - which is possibly a touch far north). Bath as well.