If you want a particular pet, go kill the monster of the same type and collect 3 DNA as Spunny mentioned. You will get a random quality pet by doing that up to Epic quality but if you want guaranteed epic quality you can buy a DNA Booster from the shop.
You can also get specific pets during events like Easter, Halloween and Christmas by collecting summoning pieces.
There are also a few pets on the battlefield exchange vendor, but it would take a while to get those.