To tame a pet is easy:
First you buy a whip and then head out and find a mob that has two yellow bars, one notched under its name. You have to whip it until the notched yellow bar hits zero without the solid health bar doing the same(mob dying). Once you do that the mob is tamed and you need to get nutrio bars(sweetstuff and fruit refined) and feed your pet and then you can train it!

Pets can also be trained but in order to keep gaining experience they need to trust you and be fed.

At the moment the taming level only goes to level 3 and not any higher, though Mindark has *promised* more things we can do with our pets. Whether you believe that or not is up to you.