A good number of the pets I bought had the tests done on them, and I got one myself, but I can't seem to figure out where to get them. Since it seems like that's really useful/necessary for selling pets, it would be nice to get them!

You get a one-time use genetic test when you first join the game. In the past, they were occasionally given out as prizes, but that has stopped for quite a while now.

The unlimited use test can be bought from the diamond shop, so you'll need diamonds to buy that. You can get diamonds by spending real money, or using gold to buy them from other players in the currency exchange.

If you don't have a test kit, you can also ask other players to test the pets for you. There are players who offer free testing services in the breeding forum.