Various sections of the Southwest Coast Path. You don't specify where you need the public transport from/to. Assuming rail you can join/leave at Poole, Weymouth, Exmouth, Dawlish, Teignmouth, Torbay, Plymouth, Par, Truro for Falmouth, and on down to Penzance. Fewer Rail links on the north coast, St Ives, Barnstaple but a nice link using the Somerset steam railway (with short bus link) to the main line at Taunton. There are pretty good buses all round the SWCP too.