If you don't already know the basics, such as renaming, feeding, or commanding pets, you should go back and read that section first.
When you work on your taming skill, you want to try and tame animals that are not too difficult and not too easy for you, since you can only gain skill on successes. The ideal animals to tame are about 20 points below your current skill level, but taming a mixture of anything from 10 to 45 below is the best (this page will generate a complete list of what to tame for your skill). Unless you locked your animal lore (if you've done this, you should unlock it now), it will go up passively while you tame.
You must always release what you tame, unless you are keeping it as a pet. Every player is limited to 5 control slots. If you've reached your maximum control slots, you will not be able to tame any more animals. Even if this restriction were not in place, you should be releasing your tames anyway. Other tamers can tame them for skill, warriors can fight them to work on their melee skills, or tailors can collect their hides to make leather armor.
You cannot tame the same creature over and over for skill gain. If you stay in a relatively small area, you will find that you've already tamed all of the creatures there. New ones won't spawn until the old ones go away. This usually takes quite some time by itself, but you can speed up the fresh animal spawn by killing your tames. Animals can only be retamed so many times before no one can tame them at all, and they, too, should be destroyed.
Adventuring isn't just for warriors and mages. So maybe you can't tame dragons just yet, but the durability of some of the medium skill level animals might just surprise you!
The most popular mid-level pet is the grizzly bear and should make a suitable companion for quite some time. Polar bears are relatively the same as grizzlies and are easier to tame, but you might have some trouble aquiring one unless you can find a way to get it off of Arctic Isle. Giant spiders also make good mid-level pets. A little bit further down the line, the awesome frenzied ostard (featured at the right) becomes available. All of these creatures use pack instinct. If you use pets with the same kind of pack instinct, they will do a little more damage than if they were paired up with other pets with a different type of pack instinct or no pack instinct at all.
As you adventure with your pet, you might find that it is getting stronger as well as improving its skills when you use animal lore on it. We call this training. A well trained pet is much stronger than a wild version of the pet. They learn their skills in the same way that us players do. Pets will raise their skills faster if they fight against creatures with similar skill levels (tactics vs tactics, wrestling vs wrestling, magery vs resist). Fighting creatures way beyond your pet's current skill levels will do little to train it.
Many of the higher level tamables have special quirks and abilities. There are a few that fall into the fire breathers category that require a great deal of care and patience. A few such as unicorns, ki-rins, and swamp dragons have very unique abilities and restrictions. We've already familiarized ourselves with the pack instinct ability. The tameable chart has a complete listing of all of the tamables with individual pages detailing a particular pet's special abilities and nature.
That should basically get you started on your journey down the path of the tamer. There are many articles throughout this site that go into more detail of things that were brushed over in this guide as well as more advanced concepts that were skipped. Good luck!