When Does The Airport Open?
If you go down to the most southern point of your island, you will come across a dock that houses a building on it. This building is the airport for the island, which is run by Orville of Dodo Airlines. You can go down to here on day one, but cannot do anything.
Instead, you will have to wait until the second real day comes along in the game. This means you have to do the initial quests that Tom Nook gives you that culminates with making a campfire and you being able to go to your tent and go to sleep.
When you wake up, this will begin the first actual day where time flows naturally with real time. You must complete any of the tasks around the island that you can, but the airport is still off limits.
What you must do is continue to play until you get to the actual next day, which resets at 5 am. At this time, Tom Nook will give a morning announcement to the island. On top of that, the airport will not be open for you to use, though you will need a Nook Miles Ticket to fly to a random island.
If you have friends that have open islands online or locally, you can visit their islands right away. Just speak with Orville inside the airport and fly away.