The nuts and bolts of Taming.
Locating tameable creatures.

Tameable creatures are identified by a heart icon at the left of the name plate that becomes visible when the creature is targeted, i.e. when your Avatar look directly at it or hover the cursor over it.
If the creature name plate lacks a heart icon the creature is not tameable.
Note that a tameable creature is neither tamed nor docile, it is still essentially wild and will attack you if you get to close to it or otherwise triggers its wild bad mood.
The Taming process

The first step in taming a tameable creature is to strengthen its respect for you. When using a tool suitable for taming, e.g. a whip the creatures name plate will display two new metric bars, Spirit and Respect.
When using the tool the Spirit lowers while the Respect increases.
Note that the creature also takes damage and since you don’t want to kill it you have to wait for its health to regenerate enough so that it can withstand the next use of the tool.
During the regeneration time the Spirit will increase while the Respect diminishes.
Staying alert and vigilant is key to be able to attempt to tame. Your Avatars taming related skills will also affect the parameters of the taming process.
To be able to perform a taming attempt on the creature the Spirit bar has to go below its threshold line while the Respect bar at the same time is above its threshold line. At this point the text “Tame Now” is shown next to the Spirit and Respect bars.

The “Attempt Tame” action can be found in the Action Book (default key ‘Y’). It is recommended that you map the action to your keyboard for easier use. You can also use the right mouse button if in Aim Mode.

The taming attempt is not guaranteed to succeed. If the attempt fails the creature will regain some of its Spirit and lose some Respect towards you. As long as the creature is alive you can continue the process and do more attempts.
Your new Pet.

As soon as you successfully tamed a creature it will become an asset in your inventory in the form of a pet statue. Right click the statue to bring up its context menu. In the context menu select “Summon” to have it join your side and start following your Avatar around.
To put the Pet back in your inventory simply right-click it and select "Dismiss Pet".
When in the inventory/storage the pets are in stasis and do not require any extra energy or care.
The Pet UI.

Once a pet is summoned a Pet UI will become visible on the screen. The UI consists of a top name and information part and four status bars below it.
XP bar. The XP bar shows the experience level percentage of the pet. As you play with, do tricks and take care of the pet new levels will be unlocked. The bar shows the percentage until you reach the next level.
Affection bar. Each pet will gain affection towards you based on how well you take care of it. Building a deep connection by treating the pet well will give bonus experience and possibly other good things.
Mood bar. Tamed pets have moods too. Play, feed and cuddle to keep it happy!
Energy bar. Energy determines how well fed a pet is. Right click food like Nutri bars in your inventory to give it to your pet using the “Feed pet” option. How well fed a pet is will affect its mood. Letting the pet starve will eventually make it grumpy.
Focus. The focus value will determine how much of your tricks will be gained in experience, affection and mood. Focus will regain over time but much slower if in inventory. If you push your pet too hard the benefit will be lower.
Leveling and tricks
When the pet gains experience and the XP bar reaches 100% it will earn a new level. Reaching a new level could mean that a new trick or other abilities becomes unlocked. The effort needed to level up different pets differs as do the features that are unlocked.
To tell the pet to do a trick, bring up the context menu on the pet or press the portrait on the Pet Dashboard UI. The currently available tricks are found under the “Tricks” submenu.
Tricks will both give your avatar skills and affect the pets experience, affection and mood values. Performing tricks will also consume some energy. Pets will not be able to perform tricks when its mood is “sad”.
Profession level for tricks
Some tricks needs a certain profession level in the Pet Handler profession to be performed. Even if the pet reaches a new level and thereby learns a new trick, you might not be able to tell the pet to perform it. By continuing to increase the skills related to the Pet Handler profession by other activities you will eventually be able to perform the tricks.
Pets and trading.
Pets cannot be sold on the auction but you can trade it to other avatars in private trade. Beware that the pet will lose all affection when it changes owner, this includes the affection towards you if it were to be given back.
Selling a pet to the trade terminal will only get you the PED-value of its remaining energy.
Tools for taking care of pets
Several tools can be used to let the pet gain mood and affection. E.g. brushes can be used to tighten the bonds between master and pet and increase the pets stats and level.