The South Downs Way (SDW) starts in Winchester (train station) and follows the ridge of the South Downs to the sea at Eastbourne (train station).
The entire route is a byeway, so cycling is legal. It is completely traffic free, along hard surface tracks, or hard flat grass. Most of the route is along fairly flat ridges, but there are steep descents and ascents where the ridge is crossed by river valleys.
Towards the end of the route in Alfriston, where the SDW spits with 2 routes to the sea at Eastbourne. The northern route is the official route for cyclists, but you could take a quiet lane on the east side of the Cuckmere river south to the sea, and cycle along the grassy Seven Sisters cliffs (the southern route - more dramatic, but much more strenuous). Once at the end, don't cycle along the sea front promenade to Eastbourne - they fine people who do this.