did you check the quest archive?

Go through the entire New Light quest line. Make sure you claim an complete all available starter quests.

Hi, after you do some of " A Gaurdian Rises " the first new light quest; you should get another quest called... Riding The Storm That unlocks your arc subclass, after that your other classes are available. When I say available though you may have to pay in glimmer to unlock a lot of the fragments & grenades etc... If that still the case just a tip... 1) I would unlock void first because it's easier to survive 2) unlock the fragments that give you stats first... like + 10 resilance, recovery etc... If you have questions feel free to ask. Cheers

You can talk to the ikora Rey at the tower perhaps?

They should already be there as options. If you open the subclass menu, there's an option to change your super, e.g. Burning Maul instead of Hammer of Sol.