When you get the hatchling it is L35. It needs to be L55 to do a quest to become a Strider. The strider is one of the two only mounts in game.
A strider can only become a Wyvern (temporarily) when mounted by a castle Lord at a specific location. (There is additional consumables when doing this.) Wyverns are Flying mounts.
You can buy a Strider at any time you want if you have the cash. (I never see them for less than 20M though.) If you are a lower level than the strider, there is a movement penalty.
Until you can do one of the above, just buy haste pots and run or use the Gatekeepers.