In order to get a Plane, you need to buy an Aerodrome at the Market.
Aerodrome costs 250K coins and is available from level 35.

Aerodrome construction stages:
1. 20 boards + 7 veneer + 7 beam
2. 20 bricks + 3 pipes + 1 buffet
3. 3 slate + 3 windows + 1 plaid throw

The Plane is fueled by Kerosene.
Kerosene can be made in the Oil Refinery (level 35, 3500K coins) from the Oil extracted in expeditions or bought at the Market.
The amount of Kerosene required depends on the distance of the flight and the weight of the cargo and equipment inside the sled. 

Cargo capacity of the Plane equals to the capacity of your sled.
Sled upgrades work in a particular manner when you travel by Plane.

Equipment fastener makes the equipment have zero weight, thus making trips (no matter how far) without any other cargo cost you only 1 Kerosene.
Falling back works the way it should, but never expires (the days are not counting).
Reinforcer gives you the extra cargo, but doesn't affect your speed.
Navigator, Skids Grease and Soft Harness do not work at all and do not expire.

Useful tip: Use your plane to travel without any cargo TO locations, and use the SLED to bring your treasures back.