This page is for the average trainer, collecting pets, interested in training a few favorites to Adult and maybe beyond, and perhaps dabble in creating a hybrid.
Relating Basic Pet stats to Training.
Pedigree
This is the first of two numbers shown over your pet's head as it walks/flies around, to the right of the double helix under their name. It is the sum of the "rarity" of all Talents and Derby Abilities your pet can learn.
The number in parenthesis next to the pedigree is the sum of the rarity of all Talents and Derby Abilities your pet already knows.
Stats and Experience
Every pet has stats for Strength, Intellect, Agility, Will, and Power, and an experience bar. The pet's stats will improve and experience will be gained based on the games you play, how well you score in those games, and the snacks you feed your pet. The Stats are very important for derby racers, but may be less important in battle depending on what Talents your pet learns. If your pet does not learn a Talent which uses a stat, that stat does nothing in battle, so there is no point maxing it.
The only way to increase a Pet's power attribute is through eating snacks.
Level
Your pet learns a new Talent and Derby Ability with each level, so leveling up is important. However, your pet's level does not improve its stats, its max stats, or the effectiveness of any of its Talents or Derby Abilities. A new level is reached when the experiance bar is full. There are five levels, Baby (starting level), Teen, Adult, Ancient, and Epic.
Level also determines the cost in pet energy to train your pet. Babies train quickly, for only 2 energy per game. Each level adds 2 more, for a total of 8 energy to train an Ancient. Epic no longer adds 2 to the training cost, so Epic pets now train for 8 pet energy per game.
Talents
Each time your pet gains a level, it will learn a new Talent (and Derby Ability). There are many kinds of Talents, and they all do something different based on their descriptions. All pets may gain a total of 4 Talents.
Player-Boosting Talents
Talents which boost your wizard's stats in a duel are based on three of your pet's stats. Which three stats boost the Talent you can see by mousing over the stats on your pet's page. Accuracy-boosting Talents, for example, are based on your pet's Intellect, Agility, and Power. Training up these abilities will improve the boost that Talent provides.
Pip Boost is a rare exception to this rule, available to school-given pets acquired before the pet update. Pip Boost is not based on your pet's stats, so they will neither hurt it nor improve it.
Your pet's Power stat affects all boosting Talents (except Pip Boost), but to a lesser degree than the other two stats.
Pet-Boosting ("Selfish") Talents
Many Talents boost your pet's max stats. These make for a less-rounded pet overall, with fewer player boosts in battle, but can be useful to wizards looking for a very specialized pet. For instance, if you have a pet which already boosts your damage and that's all you want it to do, learning a Talent which raises the max Strength, Will, or Power of your pet will allow it to boost your damage even more.
Unfortunately, these have earned the title "selfish" Talents for two reasons. First, if your pet has a Strength-boosting Talent but never learns any player-boosting Talents which use Strength, the Strength boost is wasted. Second, if you train a pet all the way to epic and all it learns are pet-boosting Talents, it will have very high max stats and no Talents which use them. You have trained your pet all the way to Epic and all it's good for in battle is a decoration.
It's not exactly BASIC information, but good to explain here that pet-boosting Talents work whether you see a difference or not. The game rounds values to nice whole numbers for us to see, but it's actually keeping track of fractions and fractions of fractions. Weakness, for instance, takes a little more than 25% off an attack. Furthermore, not all of the formula agree. Anyone who has closely watched the growth of their Pip O'Plenty pet will tell you their pet's page shows an increase in pip chance long before their player's page does. If your Pip O'Plenty pet learns a Strength-boosting Talent at Epic, further training to max out it's now-higher Strength will not be wasted. It will still increase your wizard's pip chance, even if the increase isn't big enough to change the percentage displayed on your character page (i.e. if the change is less than 1%).
Pedigree/Talent "Rarity" Undefined
The game does not explain what determines the rarity of a Talent. Popular belief is that a Talent's rarity determines how likely the pet is to learn that Talent when it levels up. For instance, a rare Talent would be much more likely to manifest than an ultra-rare, which would be more likely than an epic.
Based on lots of observation and training hordes of different breeds of pets, I think Talent rarity is instead an indication of how many pets have access to that Talent. For instance, a rare Talent could be learned naturally by many different breeds of pets, while an epic Talent would only be available to a few.
More importantly, when choosing your pet, don't worry about the pedigree. The Talents a pet can learn are much more important, and most of the useful Talents in battle are merely rare.
Favorite Snacks
These are not shown on your pet's page, nor will the game keep track of them for you. You'll have to do that yourself, but the pet pages here in the Wiki can certainly help. Your pet's favorite snacks are very good to know. "Liked" snacks will give your pet +1 to Power and +1 to experience with each feeding. "Loved" snacks will give +2 to Power and experience.
Which snacks your pet likes and loves are determined by its breed. Every Piggle has the same favorites as every other Piggle, etc.
Pets will always at least like snacks in their own school. They will also at least like at least one generic type of snack such as veggies, fruits, or spicy... It could be that some snacks belong to more than one generic type. These types are not ever specified in the game but there is "evidence" that this "typing" is coded into the snacks in the July 2010 update: The quote under snacks is: "The humble Tomato will remain in the veggie aisle" the assumption is that it will do so even though a tomato is a fruit. If you find a type of snack a pet likes, and then you find a snack of that type that is also in it's school, it is highly likely to be a loved snack.
Training Efficiently
There is no right or wrong way to train your pet. But, when faced with the prospect of training several pets to find the best, or training a pet to Epic, it's best to have a plan. Epic can take as little as one week or as much as three. Planning ahead can save you lots of time and effort.
Timeline
High level players can train a pet to Teen within minutes of hatching. Lower level players, or players with pet energy empty when their egg hatches, can still have a Teen within eleven hours. Adult will take you another day with an average +3 experience per snack. Ancient will take two days with the best snacks (+6 experience). Epic will take 5.5 days with these same snacks. A grandmaster with a good plan can have an Epic pet in about nine days. High level players who are not yet grandmasters (about level 35 or 40), gaining a level a day, can use the pet energy refill when they level up to their advantage, getting to Epic in just under a week.
Pet Game Practice
No one is good at every pet game their first try. But you need max points from them if you don't want to add lots of time to your training. Practice, practice, practice!
If you simply can't get the hang of a particular game, other games offer "split" versions which can help. For instance, if you really stink at the maze (which boosts Intellect +4) you can play the dance game on the Dragonspyre setting (which boosts Intellect +2, Agility +2). You can still get a pet to max stats long before it's Epic by only playing three of the pet games, or even two of the games if the "split" version each game boosts different stats.
Snacks, When and What to Feed
Your newly-hatched pet offers an opportunity it will never have again: you can double your pet energy as experience. You can never double the effectiveness of your snacks. Not feeding your babies snacks is an excellent strategy. Feeding at this stage offers a minimal boost compared to how quickly they earn experience without snacks, and it saves your snacks for later levels (or possibly for later pets), when pet energy helps little and the snacks are really needed. A key exception to this if if your wanting to make a teen or adult derby racer. You will want as much power as you can get and snacks are the only way to increase power.
Teens still have an equal swap for pet energy to experience, and only need 250 experience to reach Adult. Feeding at this stage is fine, but it's a good idea to start with your smallest snacks (and a good time to get those +1 snacks out of your inventory). Save the bigger snacks and bigger boosts for later levels.
Adult and Ancient pets gain experience very slowly, and they require a lot more of it to level up. This is the best time to bring out your biggest snacks. If you haven't started crafting, pet snacks are a good reason to start. The Atheneum and Ravenscar offer three cereals each. These are the best pet snacks currently available without crowns, +6 to experience, and most pets like one or more of the cereals for an even better experience boost. These will get your Adult to Ancient after only 50 games (and 50 snacks), or about two days. Ancient pets will require 100 cereals to reach Epic, unless you find a cereal your pet likes, in which case it will need 91 of them.
Knowing When to Quit
Your first pet probably won't be perfect for what you want. If you face the prospect of training several pets to find one with Talents you can use, setting a "quitting level" for yourself will save snacks and pet energy for other pets and better prospects. A baby has a 40% chance of learning a particular Talent by Epic. A Teen which hasn't yet learned that Talent only has a 33% chance of learning that Talent by Epic, Adult has 25% chance, and Ancient has a 14% chance of still learning that Talent after the 5+ days it takes you to level it up. :P
If you're looking for one or two specific Talents, Adult is an excellent time to quit. A Teen still has one in three odds, much better than the one in four an Adult offers, and training to Adult only takes two days.
If you have several pets in your inventory or you're farming a boss enough to get an egg or two every day, training only to Teen may be best for you. Or you may have a truly rare pet like the Ianthine Spectre and decide to train it to Epic no matter what Talents it learns.
Ultimately, when to quit is a personal decision only you can make.
Advanced Customization
The ultimate challenge is in Creating a custom pet to have particular talents, stats, looks, and card. Check out the Creating a custom pet page for details and an example.