Fast travel is a common feature in many RPG games, and such is the case for Biomutant. However, the mechanic takes a strange turn as you will not be able to fast travel in the usual way, i.e., an option in the menu. Instead, you will need to find various flag posts with a yellow flag and mark them - this is exactly how it sounds.
After this, you will be able to use that flag post for fast travel and will be able to instantly fast travel from one to another. That said, you can find these posts all over the game world, so just explore, and you will find one sooner or later. But if you prefer something instead of fast travel, you can always get a mount.
Related: How to get Psi Points in Biomutant
In short, to get a mount you will need to either buy them or get a fruit and feed it to one of the various animals in the game. Getting a fruit called a pip and then feeding it to an animal or mount will tame them, allowing you to ride them into battle. This is an excellent alternative for those who rather explore but move at a faster rate.