I couldn’t find any quick information on who it is that starts the questline to tame my first pet, so I’m making it. I remembered that for my dwarf hunter, the trainer was not in Ironforge, so I figured that the trainer for Blood elves would not be in Silvermoon. Wrong.
Upon reaching level 10, hunters gain the ability to tame their pets, after completing a questline. The questline starts with Oninath in Silvermoon (84,26). He sends you to Farstrider Retreat to speak with Lieutenant Dawnrunner (60,63).
The kind Lieutenant sends you off to tame a Crazed Dragonhawk. I just ran out the door to the North and tamed one right away. Nice and simple. Run back and complete the quest to pick up a quest to tame an Elder Springpaw. I spotted a level 8 to the West of the Retreat and tamed it. The final taming leg is for a Mistbat. This one took some running. Follow the Dead Scar all the way to the border on the river. Staying on the East side, there are some mistbats on the other shore. I had to settle for a level 10 and tamed it (barely) to finish the questline.
I ran back to the Retreat and went on back to Silvermoon to do some pet training. When I went back out into the world, I tamed a level 9 Elder Springpaw and we can hunt without stopping. He has more HP than me and does about 11dps already.
Update with screenshots:
Due to the vast number of comments on this post and people e-mailing saying that I’m wrong, I re-rolled a hunter again and here is exactly what happens when you hit 10 as a Blood Elf hunter.
He offers me the quest in addition to his normal hunter trainer abilities.
Here is what the quest says to do: