To get into Eastern Island theres just a few options:
Flight travel on LAN from Santiago de Chile in Chile or traveling from Lima, Perú. This last flight is just some days of the week so you have to know exactly the date of your trip and see if you can make it.
From Chile or Perú is a 5 hours trip so you just sit and enjoy the flight (boeing 767)
Once youre there you can find a lot of options to stay and to do on the island but in this blog you’ll find just a few advices based on my trip.
Hope it will be helpful.