I was born and brought up just outside Chi, so I appreciate your difficulty in getting to the woodland on the Downs from Selsey without a car. However, you do have some excellent coastline to explore, I'd start there.
From where you are I'd also suggest some form of boat too (with appropriate training, blah blah). There is a very active group for the Solent area within the Dinghy Cruising Association who frequently meet up in Chi harbour or sail across to the Isle of Wight. I'm not talking about huge yachts, but dinghies of ten to 16 feet that can be bought for relatively little money. In the past I've launched a canoe at Climping and been up to Pallingham Quay on the Arun and back. Last year a mate and I paddled from Chi canal basin to Dell Quay via canal and Fishbourne Creek. I realise that a car is going to make using a canoe much easier, but if you have deep pockets then folding boats are available.
Don't rule out the distances that can be travelled by push bike either. While it's been twenty years since I left Sussex, I found even then that a push bike could be quicker than a car for many trips because of the volume of traffic.
Get an OS map and start looking for bits of woodland nearby, I'm sure there'll be more than you think. You've a lot on your doorstep, get out and enjoy it!