I'm not finding anywhere in the rules where it says "This is how you start a quest" or "this is when you draw more of them." Do I just say "I'm doing a quest?" Do I have to be in a specific location? And again, when do we draw more?
HOW do you do quests?
Quests are typically found in Wasteland and Settlement encounters. An encounter result would say something like: "Stage card X" or "Add card Y"
When staging a card you place it face up in the staging area after which anyone can attempt to achieve it on a first come first serve basis.
When adding a card you check that cards back, then draw cards (without looking at their fronts) from the corresponding pile equal to the number of players, shuffle the drawn card(s) with the card being added, then place the (number of players+1) cards on top of the same deck.
After a card has been added it will eventually be revealed (and possibly staged) with a future encounter at an appropriate location.
Note what Fnoffen is saying about Stage and Add . Those are two very different mechanics - Staging means it's a full-on quest that adds to the board, and Adding puts a new card into one of the Encounter decks.
But as for your question about "doing" a quest, sometimes you do indeed say that you're questing as one of the actions on your turn, but many times you don't even have to do that. If there's a quest card where you have to travel to a specific location and, let's say, discard a follower, then once you're at that location, you must use one of your actions to discard the follower and get your reward. These types of quests will also have a Quest Icon on the card - it's the little circle-star icon that looks kinda like a vault door (Page 11 of Rules Reference Guide).
On the other hand, if there's a quest whose sole requirement is to kill a Raider Psycho, then you get the reward for the quest as soon as you kill any NPC whose name perfectly matches "Raider Psycho," EVEN if that NPC initiates the combat when you move into its space. No extra action needs to be spend to acquire the prize - you simply have to be the first player to kill the mob.