After Pet Master is up to level 5, Riding is unlocked, but requires one skill point to be enabled. Once done, all animals listed as "Ridable" on their cage description can be mounted.
When riding a pet, your pet will become thirsty. The hydration level of your pet is represented by the water drops under its name on the left side of the screen. Once your pet's hydration level is depleted, you will be unable to ride it. You can replenish your pet's hydration level by standing in a pool of water (the same as you would when filling a flask).
You can increase the bonus speed as you put more skillpoints into Riding.