First things first, make sure you have bought the Battle for Azeroth expansion pack. You will obviously not be able to begin playing it without this. The next step is to make sure your character is level 110. I think most people have done these two, but I just needed to throw it out there.
Completing the requirements to begin BfA contentBefore you can physically go to either Kul Tiras or Zandalar, you will need to have completed the Battle for Lordaeron scenario on the character you want to unlock BfA content with. This scenario begins with a pop-up quest for all eligible characters, and you do the entire thing in Stormwind Docks or in Orgrimmar (depending on your faction of course). This scenario is pretty quick, and any level 110 character can get the quest to do it after BfA has launched (you needed to do the Darkshore quest line prior to BfA launch before you could do this).
Once you have completed the Battle for Lordaeron scenario, a new pop-up quest will take you to meet Magni Bronzebeard in Silithus, which will give you your Artifact necklace. Just follow this small quest chain, and your character will soon be transported to Kul Tiras if you play Alliance, or Zandalar if you play Horde.
If you do not get the quest to go to Silithus to get your Artifact, relog your character. If you still don't get it (but are sure that you did the scenario), go to Stormwind or Orgrimmar where the quest will be forced to pop up for all eligible characters.