There are three ways in which you can get a mount in Biomutant. The first is to tame them using a fruit called pips, the second is to purchase them from various vendors in the game, and the third is to create them through quests.
That said, if you want to tame a mount, you will first need to locate one. Luckily you will get your first mount when you capture your first outpost in a scripted event.
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During this event, you will learn that you can get pip fruit by dodging underneath bushes with red fruit on them. Doing so will give you a pip fruit which you can then use to feed an animal you are trying to tame. Once you got a fruit, simply find an animal walking around and walk up and feed the fruit to them to tame them.
If you want to buy mounts, you will need to locate one of the various vendors in the game. You can find these vendors in outposts and can buy mounts from a special mount vendor. Now the last way you can get a mount is to find and complete certain quests in the game, so keep playing, and sooner or later, you will get an awesome mount.