If you do not already have the pet you want: Kill the type of pet you want until you get an egg. After you have obtained the egg, go to Astraea's Bastion, and put the egg into the stable. Wait until it hatches. Click add, then click yes.
If you have the pet you want: Go to your stable, click on it, and replace it as your active pet.
If your pet has died: Pick up the blood dropped by it. Return to Astraea's Bastion, go to the stable, talk to the Sigil with a red eye, and then pay for it to be reborn. Then follow the steps above.
I think I liked the Castaway 1 pets better because after you hatched the eggs, the pets would be level 1. I like that better because you actually have to train the pets, instead of getting the pet at the level that you killed the monster at.