Brighton is not much more than an hour away. Camber Sands takes a little longer, but actually has sand (nothing else, but sand nevertheless).
Kent is full of nice spots to go. My recommendations are:
Down House which is in a nice little village with Charles Darwin's House.
High Rock for a nice walk with surprising rock formations and a steam railway running past the pub.
Closer to home are Chislehurst Caves which are another geological surprise so close to London.
And there are lots of nice woods: Petts Wood, Keston Ponds, and Sydenham Woods.