The rules for this were very confusing for me. If I have 1 health at the start of my upkeep, and draw the normal card that brings me down to 0 health because someone else has a card that deals 1 damage for every card drawn, do I leave the game at that moment, or can I still do things before I die?

When you hit 0 is when state based effects remove you from the game.

Yeah, as soon as the effect that put you at 0 has finished putting you at 0. Before the game moves to check the next thing that's happening, it checks for "state based effects", sees you're at 0 and makes you dead. Unless you're already a Lich.

well hold on, though. the card in question (presumably underworld dreams) is a triggered ability, which means it doesn't happen literally as soon as you draw a card. there is a delay between when you draw and when you take damage, and in that time you can cast instants in response to the ability, including whatever you drew. the other posters are correct that you lose as soon as you hit 0 (or, at least, close enough for these purposes) but in the specific case described, you have a window before you actually die, not because you're surviving with 0 health but because you're not at 0 health yet.