Left click a pet you have tamed with the Pet Wand to convert them.
Left click the pet while holding the Pet Wand attempts a level up.
Right click the pet while sneaking to open the inventory/skill tree.
Right click the pet with any other item to make them sit/stand.
Right click otherwise with the Mountable skill and a saddle in the saddle slot to mount the pet.
Dogs can eat beef and pork.
Cats can eat chicken and fish.
Pigs can eat carrots.
Slimes can eat cookies and raw potatos.
The Pet Wand is crafted with a torch made of bones and gold nuggets. You get one when you log into a world for the first time though.
Pet claws are found as dungeon loot, just like saddles. However, in the config file you can enable a recipe for them, and for saddles.