Most pet guides are outdated at the moment.
Loyalty: There IS no loyalty. Instead of slowly gaining affection for you as you adventure with (and feed, feed, feed) your pet, it is as friendly as it is going to get towards you as soon as it is tamed.
Training Points: Likewise, there are no longer training points that are build up based on loyalty level and pet level. You do NOT need to go out and tame other pets to learn abilities, your pet will learn all basic abilities (cower, growl, focus dump, special) as it levels up.
Talent Points: A new addition, pet talent points let you customize your pet's abilities and stats to a certain extent. They work just like your own talent points with the exception that each type of pet has access to only one tree. Your pet will start getting points at level 20, with more every couple of levels (not every level like you do) and at level 80 will have either 16 or 20 points to spend, depending on your own talents. For more information on the pet talent trees, see
Petopia.
Pet Trainer NPC: A little obsolete--they no longer teach you new abilities to teach to your pet. They still will let you pay a small fee to retrain your pet, however, starting at 10 silver or so--the only difference is that now it resets the pet's talent points instead of resetting its training points. Also note that when you buy a respec for yourself a respec for your pet is included free of charge.