I can't see any story missions on Destiny and I have not beat the game yet

When I go to orbit, I go to destination and there isn't the little purple triangle to tell me where the next mission is, the only way I can do a mission is if my friend invites me to their fireteam and he picks the mission. Is their any way to solve this problem?

There is a kiosk in the Tower in the Vanguard section (down the stairs past the crucible section) on the right. It is the abandoned Quests kiosk. Perhaps your quests were discarded? Either way you should be able to pick and mission that you have previously played. Can you upload a screenshot or picture of your director (mission select) screen?

The missions you are on either currently on are interfering with getting new missions, or you've discarded a quest somewhere. Check your quest log-then check your orbit locations. If nothing is there, go do all the quests in your quest log.