Locate a quest NPC; they have a blue question mark symbol above their head.
To start a quest, simply bump into the NPC.
Follow the instructions that appear on the screen, that's it.
(You can also choose to cancel starting the quest by selecting “No” in the blue prompt text in the right-hand side of the screen, however you won't be able to pick it back up until after a cooldown period of approx. 10-15 minutes)