Don't let the word "PVP" scare you off, the advice in those threads apply to anyone who wants to get into hatching. One gives secrets about how pet talents are inherited, for example, and another suggests ways to earn all the gold and pet snacks you're going to need.
This is not a fast process. Patience, persistence and keeping notes of your hatches are truly necessities. Also, start out with the best, most solid pets you can. Having strong stats and talents to start with can cut down the number of hatches you have to do by quite a lot.
I am terribly new at the whole “hatching” process. Will someone please guide me through how to determine what to do? Thanks!
-Destiny Duskstrider Balance lvl 71
Basics: -You can hatch any two adult pets. So you can hatch two of your own adult pets or hatch your adult pet with another player's adult pet. -Some pet combinations produce hybrid pets. Hybrid pets often look different and have different abilities/talents. Popular hybrid pets are mixed between School-Only pets. For example, Sabertooth (Balance-Only) hatched with Avenging Fossil (Death-Only) can produce a Forlorn Sabertooth. -Other pet combinations produce a pet that looks like one of their parents. These pets can have a combination of talents from each parent.
More Complicated: There are some theories that there are "dominant" pets. For example you hatch Pet A with Pet B. Pet A is a dominant pet therefore increasing your chances of hatching a Pet A.
Wizard101 Wiki is an amazing place to research more about hybrid pets and hatching so be sure to look at their Pet section.