I finished up my Sunsinger quest(where you had to get 25 kills, 50 orbs, and then a strike) and was only rewarded with a bond when I thought you would get a bond and an emblem? If there was a PvP section of this quest, then shouldn't it have unlocked after I had done the PvE section(which I just did)?
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Check the abandoned quests kiosk. There's a possibility the first quest is there somewhere. You get prompted for the subclass quests after you finish the menial class quests your Vanguard gives you. I think it's level up to 5,10,20, etc.. The Hunter's first subclass quest was throwing knife headshots in the Crucible. Just as an example. Good luck.