How do I get rid of my pet?

Same as the Title.

You can use the new menagerie and talk to Emryd the trainer to release the pet into the menagerie, or you can look for the pet tamers. One is just outside of Ascalon. Check the wiki for other locations.

Also one outside Shing Jea Monat.

Well... The more you pay the faster the "ridding" is done. Painless "jobs" are harder to pull off and I need a little more dough, if ya know what I mean. For a quick job is 1k, for a painless it's 2k, I don't have all day. If you don't like it, talk to the Zaishen, they take care of "pets" at slow pace of collecting coins.

A serious note, talk to a tamer found outside of Eye of the north over by the garden next to the outpost, Shing Jea to the left of the wall going out, Ascalon City right out front, somewhere outside of Spammadam in the training area. At the Battle Isles, visit the new pet area thanks to the update.

Go to a pet tamer.