Hiya!

There's no real level requirement per se - you get pets from quests, items and trophies. The easiest pet to get is a Baby Bluebell - for that, just complete all the quests at Theo's farm (don't forget returning the lost baby bluebell in the far east of the forest!).

If you see Lint in Saliko forest, it can drop a huge carrot, which lets you pick any Fliffy as a pet - so if you do find Lint, it's a good idea to fight it. ;3

Some quests and trophies have level requirements (for example, you can get a dungeon trophy that will award you a spider hatchling, and the dungeon itself has a minimum level requirement) - but there is no direct restriction. As soon as you can get the quest, or the item, or the trophy, you can get the pet.

Glad you like the invisibility skill! Mages have a lot of very neat abilities!