First, you have to pick a pet. This may sound easy, but I find it's more difficult than it seems. Do you want a cute pet? How about a pet that gives you a myth blade card? Or do you really want a pet that will give you a healing card like Satyr? You have to pick one. Just one. Pick some pet that you reaaaaaaally like, because you'll be working with it for a while. For me it was a Red Gobbler pet. They are so adorable. A little fidgety ball of goodness. Tip: most crowns shop pets have a hatching timer of only 5 minutes, which means you can start training right away. Other pets have hatching timers of 18 hours or longer! Choose your pet wisely!
Second, stats: there are many different stats (you can find the biggest catalogue here at Petnome), one pet can only manifest 4 out of 10 talents. So you have to decide which 4 talents you would like in the perfect world, best case scenario. That is what you will shoot for. Let me give you a couple of examples:
This is my friend Jeremy's pet. It's what's commonly called a "SPUD" pet. It has general resistance (this version has 15%) and two healing moves that will randomly heal you in a fight. I call this pet of win.
Here's another example. This pet belongs to some dude I kept seeing in the hatchery a while back. This is a completely different kind of pet than a SPUD pet. It's much more specific, because to be able to get the full benefit you'd have to be a storm wizard, and it has no healing or other "may cast" abilities. I call this pet of owwwwwww.
If you're now drowning in all the possible awesome pet combinations, and have no idea what to pick, fear not. I recommend taking a look at what other people like in their pets. If you're an ice wizard, and you want a really wicked ice pet, take a look at what other high level wizards have on their pets. Good pet viewing places are the Arena lobby, and the Hatchery. I recommend looking at epic pets, because generally they have the stats the person was looking for. Take a picture if you have trouble remembering like me.
If you still have no idea what you want, or now you have too many ideas, go for a SPUD pet. They are a good middle ground pet.
Decided on something? Good. Now we can get into the mechanics of how to get that. My friend used the metaphor of "mixing paint" when he explained it to me, and I think that captures the process pretty well.
Okay here is a simplified example:
You love your Stray Piggle (I said it was an example!). You really want to have spritely on your Piggle. Your friend's Mander pet has spritely. I know what you're thinking. Mander pets are lame huh? Well what if the mander looked like THIS!
Ahem. So you hatch these two pets together.
What pet you get from this hatch will matter.
If you get a mander, you'll be getting a mander with slightly more "meh" talents.
And we're building a super pet! We're not going to settle for "meh"! Besides, you like piggles. You want a piggle.
So if you get a Mander you're going to toss it, and rehatch as many times as it takes to get a piggle.
Okay now you've got a piggle that has more mander-like talents. This is what you want. So you'll raise this piggle up to adult. Chances are this piggle is not going to get spritely. The odds are too low. This 2nd generation piggle still has too many "meh" talents and not enough awesome mander talents. (If you're pet does get spritely on the first go around, be happy in the thought that you are blessed with more luck than a leprechaun sitting on a horseshoe, holding a rabbit's foot that is clutching a four-leaf clover.)
Okay, so if you're normal like the rest of us, and you didn't get spritely on your first try, take your failed piggle and hatch again with your friend's mander. This keeps introducing the mander's awesome talents into your piggle, thereby increasing the chances that you'll actually manifest spritely. Simply, every piggle you get, you hatch to back to the original mander. This concentrates things; if you threw in a random Firecat you don't know what talents are being pushed aside to make room for the firecat's talents, the spritely that you're trying to get might be lost!
Essentially that is hatching. You repeat the process until you get spritely to appear and then dance around in little circles for joy.
To be continued...