The first step to getting to Southern Island is to Mix Records with Ruby or Sapphire which has a legitimite Eon Ticket. One this is completed, you should have a notice saying that they transferred the Eon Ticket to you.
It will be placed in your Key Items Pocket of your bag so you will know if you have it
Go To Lilycove City's Port and talk to the Boat Admitter. She will notice your Eon Ticket and send out the captain who will decide to take you to Southern Island
It's a basic Island with a forest cave. Go into the Forest and you'll see a Pedestal. Go Up to that and either Latios or Latias will appear depending on what one you picked after you beat the Elite Four (e.g. If you picked Latias, Latios will appear in Southern Island). It is Level 50 so Be Prepared. You only have one Shot, but you can go to Southern Island any time after it from Lilycove's Port