Does it make most sense to just start playing with setting up all the
pieces, or are there smaller "games" that one can/should start with?
It doesn't really matter the age of the person learning to play, there is no point in starting with a full set. There is just far too much to take in and make sense of. The first thing to do is to teach them how to win!
So, first teach them how king, queen and rook move and then teach the basic mates -
And I would do it in that order because it is the easiest and it is the way to introduce concepts like checkmate, king not allowed to move into check, two rooks working together, opposition, king and one rook working together, stalemate (to be avoided with KQ v K).
Next I would go on to KP v K, when you can win, when it is a draw and how to win.
Then I would go on to a game which many people start with (but there is no checkmate, so less fun IMHO) and that is 8 pawns v 8 pawns. Winner is the first to get a pawn to the other side or to "stalemate" the opponent - i.e. the opponent is the first to run out of moves.
After that I would reintroduce the kings and major pieces before finally going on to the full game.
You can see I'm working backwards here because that way the learner first learns where they should be aiming to get to and what to when they get there in stages starting from the destination.