For teaching, I have started with the 3x3 for a long time. It teaches eye vs. no eye.

It can also teach ko. It can also teach seki.
It is good for motivation if they can play games that they can win around half the time. That can mean having them play each other, but there are problems with that. For one thing, they can develop bad habits. For another, they are segregated.
Nowadays I like the capture game, played on small boards with no passes. The idea of territory emerges naturally when each player makes live groups without losing a stone. Then the player with more territory wins (except for group tax). They can move on to capture-2, capture-3, etc. I hear that up to capture-5 is OK before moving on to regular go. You can handicap the capture game by number of captures, as well as by handicap stones.

I have a DDK friend, for whom a sufficiently large handicap on a 19x19 board would be tedious for yours truly. However, the capture game on a 7x7 with him taking Black, works fine.

He doesn't win 50% of the time, but he wins often enough to keep his interest. And it is not tedious for me.
