Before you attempt to travel to Eastbourne by train at the weekend check first that there are no engineering works planned. Otherwise you will find yourself doing the best part of the journey by bus... very very irritatingly slowly as it goes round every obscure backwater, and what was a 20 min train journey becomes a 1.5 hour nightmare!
Apart from this the rail services to and from Eastbourne are normally very good. Trains go to Brighton, Lewes, Gatwick, London, Hastings etc
Buses are probably the second most convient mode of transport after cars/motorbikes. Current fair for an adult is £1.30 which is kinda a rip off. It's hiked 20p already this year and its only september . You can get travel cards etc but I don't know anything about these.
Train is the best way getting to Eastbourne from London. Cost is at GBP23 per person one way and try to avoid journeys on Weekends as there is always engineering work takeplace. I end up always changing trains and buses took long hours just to get to Eastbourne.
Train station in Eastbourne is a minute steps to Eastbourne High Street.
I don't often come to Eastbourne by bus as the train is quicker. There does seem to be a very good bus service in and around the town though - even on a Sunday, as the central shopping area is open from 10am-4pm.
Beachy Head is off the B2103, from the A259 between Eastbourne and Newhaven. Eastbourne’s city sightseeing tour bus stops at the clifftop.
You can get to Brighton or Eastbourne (either way) by double-decker bus which only cost GBP2.40 one-way. Its also another way of sightseeing in between Eastbourne and Brighton.