Once you reach level 50, you will receive a Maple Guide mission that will teach you how to pacify, tame, and upgrade pets. Maple Weeb's guide down below does a very good job explaining everything about pets. Just a warning, renaming your pet more than once will require you to spend 100 Merets.
A good map for collecting a lot of pets at once is in the Crooked Canyon map. A good map to farm potion solvents is Cave of Eternity or Red Wall. Starpyy on Reddit has also shared a good spot, which is the Abandoned Charnel House in Logger's Hill, near the top-side portal. There are 4 spawning areas for the Stirges there, and you can go in a rotation where they will respawn as you get to the next spawn location.
's Pet Video: Link