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).