Gara the Spirit Beast Taming Guide
First of, I learned of this all from eyesofthebeast.com, official link = (How to tame Gara the Spirit Beast | Eyes of the Beast)
Basic Requirements to Start:
Level 90+ Hunter
Beast Mastery Spec
The above image is Gara's location; head here first to find her. If not in BM spec, she will be unresponsive, and not have any dialogue options, but if so, when you first interact with her, you will get a simple dialogue option where she acknowledges your presence.
The next step in taming her is to simply kill a nearby rare, Mother Om'ra, and take the Shadowberry from her corpse. When returned to Gara, a new dialogue option will appear for you to interact with her, this time allowing you to get her attention.
After this, you now have to travel around Draenor and collect specific items that drop from various mobs- each in different zones. There will be one mob in each zone (Gorgrond, Talador, Arak, Nagrand, and Frostfire) that drops the items that you need, that, when returned to Gara, will be assembled onto a "Spirit Effigy" that allows you to make further progress in the questline.
List of Necessary Pieces
Crude Effigy: It drops from Drywind Bonepickers in Gorgrond.
Wooden Bowl: It drops from Gorebound Legionnaires in Talador.
Elder Incense: It drops from Sethekk Prophets in Spires of Arak.
Small Fruit: It drops from Warsong Marauders in Nagrand.
Torn Cloth: It drops from Necrophytes in Frostfire Ridge.
Ancient Bone: It drops from Thunderlord Giantslayers in Frostfire Ridge.
After assembling the Spirit Effigy from these items, return to Gara. A grave will appear on the ground that prompts you to place the effigy on it. The spirit of Mother Om'ra appears.
A giant void beast now appears, and takes both the spirit and Gara to the void.
You now need two more things to travel to the void- the Void Lantern, which drops from Shadowmoon Voidtwisters, and the Shadowmoon Voidblade- a blade stuck in the trunk of the "Great Tree Valuundira," which can only be seen if you have the Void Lantern in your possession. Its location is shown below.
You are now in the spirit realm, and Gara appears at your side as a guardian. There are a bunch of Elder Void Lords/Elder Voidcallers surrounding the tree; they are level 100. You might also need to be level 100 in order to complete this step. Once you kill enough, Xan, the Lord of Glass, appears. After killing him, you are now able to tame Gara, and keep her as a pet.
Here's the linked video from the "Eyes of the Beast" site showing off the pet.