There's a pilot's school that you have to complete as part of the plot which gives you your license, but it's not until very far into the game. But you can get your pilot's license early by sneaking into the airport and stealing a plane. The more you fly, the better you get and you'll eventually get your license.
However, the airports are locked until you get your license. To get in there's a few different ways. Some people have said you can stunt in using a bike and a packer, some have climbed onto high vehicles and climbed over the fence. I found out that if you crash into the gate on a motorbike and fall off, the gates will open when you're getting back on the bike.
When I was sneaking into San Fierro airport, I found that nearly all the planes were locked apart from the Maverick helicopter. I don't know what planes are unlocked in Los Santos at the start.
When you have your license, the airports are unlocked and you can access planes and the runways without having to sneak in over the fence.
KM.