How to get from London to Eastbourne:
It is bit different from Brighton, even though both of the towns are on the south coast, only few kilometer from each other. There are only few buses every day going from London to Eastbourne / from Eastbourne to London. But there are trains going from/to Eastbourne every 30 minutes. Trains are really expensive here in England, but if you book the ticket few days before traveling, you can find really good offers. We found £5 ticket from Eastbourne to London, we didn't book it in advance so the ticket was £7.50, but still it is not the usual price, the prices are usually 3 or 4 times higher. It took 1,5 hour from Eastbourne to London Victoria station. Check the Southern railway website.
How to get to Beachy Head and Seven Sister from Eastbourne/Brighton:
We were lucky, our host had a car, so we didn't have to take a bus. But it is really easy. You can take the £4 tourist bus from Eastbourne - it goes to Beachy Head and Seven Sisters.
Or you can take the 13X bus from Brighton. One day ticket for Brighton buses is only £3.50 if you by it by mobile phone (if you have UK number), £4.40 when buying it in a shop for example. Check all the prices, schedules and other Brighton buses here.