If you are coming from London, you can take the train from Victoria Station to Eastbourne. A same day return ticket costs 31.60 pounds. You alight in Eastbourne. This town itself is beautiful and has nice beachfront. It is worth to spend half a day here if you can spare some time. Else you can go straight to the visitor's center not far from the train station and buy a day pass for bus. This day pass costs 4.50 pounds for unlimited rides on Brighton and Hove buses. What you are interested on is the 13X tourist trail bus from Eastbourne to Brighton. You can hop onto this bus near the visitor's center or you can hop from the first stop at the beachfront after your little sightseeing detour. Take note there are limited timings for this bus route on weekdays. Only during the weekends they have the hourly sessions. So I suggest you study the timings carefully here and download the correct timetable for weekdays or weekends.
I suggest you alight at Beachy Head so you can start your hiking. You'll be hiking along the cliffs through gentle rolling hills. A casual hike until Birling Gap probably takes around 1-2 hours depending on how long are your photo stops. Take some refreshments at the lodge at Birling Gap and after that you can head North following the road until Sheep Centre and East Dean village. That should wrap up your day and from East Dean, you can take the more regular buses plying on this main road either to Brighton or back to Eastbourne for your return train back to London.
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