the quests in destiny 2 are more like background things, there arent really any "collect 10 boar tusks" type quests. it's more like you might have a quest such as "kill 20 opponents with headshots in crucible" and then as you play pvp matches you try and get headshots to fulfill this. all the activities on the planet are static/individual to your game and stay there, with the exception of public events that move around. these launch every few minutes so there is almost always one going on
Oh dear, is it PVP only? I was under the impression it was PVE and PVP ---
