Amazon Game Studios didn't make New World mounts different from each other in any way other than the looks.
That said, some players will prefer to ride a wolf or a lion, rather than the horse which is the first mount one can get so here is how you get each one of them.
Since the horse is the first mount, it is acquired immediately in the first steps of the mount quest.
Wolf mount
In order to get the wolf mount, you will need to progress the mount quest by completing two sets of races, which consist of time trials at long and short ranges, the two types being tours and rallies, respectively.
Once you've completed the first two sets, the quest will send you to speak with Valencia Iznov, who will offer a quest to save a pack of wolves from the corrupted Varangian knights. Save the wolves and your reward will be Bloodfeast, the first wolf mount.
Lion mount
In order to get the lion mount, you will need to do another two sets of races, which become incredibly tedious. Just to illustrate my point, one of the tours will send you from Brimstone Sands to Reekwater.
Once you've completed those two sets, the quest will send you to Elysian Wilds where a lioness will ask you to save her mate. Do the quest and you will be rewarded with a lion mount.
Store mounts
Alternatively, you can always just spring for the store mount skins and get them as soon as you are able to ride. Keep in mind that the "micro" part of the microtransaction concept has been lost in time as these skins can cost around $20 a piece.
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