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Simply play through the story. After the story go do the follow-up mission in the ‘The Traverler’ hub area (marked on map). This will instantly put you at level 20.
Unlock all subclasses with your first character. In Destiny 2, subclasses are obtained from finding Artifacts in random loot chests. One after level 7 and the other after level 14. The artifacts initiate small quest lines that result in the subclass being unlocked. You can farm loot chests from public events. The higher your level, the better the odds of finding the Artifacts.
Also do Shaxx’s Call ot Arms Quest. You can talk to him in the Farm hub area or ‘The Traveler’ hub area. For this quest you just need to play a few Crucible PVP matches.