MrMormon: I've had several opportunities to teach people who know little or nothing about Go. I seem to have a consistent pattern. I usually set it up somewhere like a board game club or some social event with tables, and I often get someone walking up and asking what game it is (if it's a guess, Othello - "that's about capture by sandwiching; this is about surrounding"). I say it's a game about surrounding the most territory and that it isn't just any game, but a deep one like chess.
If they want me to teach them how to play, I start by using upside-down (magnetic) stones to mark a 9x9 board and put down some stones to enclose an area (emphasizing that this is the object of the game), counting for them the number of intersections(!) of territory, and then going into the rules like this (answering questions and frequently asking if they understand):
"We take turns putting one of our stones anywhere we want, but there are some rules. There's capturing. If you have a stone here *puts down a stone* and I put stones on all of the spaces next to yours along the lines *points to the four adjacent points and puts down three of my stones to make atari*, that stone gets removed *emphasizes putting down the capturing stone and returns the captured stone to its bowl (I teach area scoring)*. Now, if you had stones connected along the lines *makes a small L shape*, I would have to put my stones on all of their free spaces *does so and removes*. Remember, diagonals don't count, so these stones *puts two stones on a diagonal* are two groups.
Also, there's a special rule for capturing *set's up a ponnuki*. Normally, this move *put's a stone inside* would be suicide since it would have no free spaces and be removed immediately *returns to bowl (I don't forbid suicide)*. But if it could capture *makes a ko pattern* a stone, that would happen instead of suicide *takes ko*. *I wait a bit to see if the person I'm teaching points out the ko.* The final rule handles a repeating situation *demonstrates the ko*. You're not allowed to recreate a board position you made (I teach SSK). If we play elsewhere, then that would be a different position and you could then recapture. That's pretty much the rules, but the rules have almost nothing to do with the strategy, so it's best to get a feel for the game by just playing."
I ask if they want some handicap stones down (I like free handicap but I usually just give them four corner pieces) and if not, who should be black and go first. I try to make few suggestions but still interact, but sometime during the game I ask if they want me to give them some suggestions (usually in the form of "Hey, what if you tried this? I have a weakness there, or you can save your group."). I eventually tell them that the game ends when we both pass, meaning that we don't think we can increase our territory or decrease the other's territory. I often work in something about Go being a game of balance, like between moving fast *shows my knight jumps (they almost always put their second stone next to their first)* and having weak stones far away from others (brought up when I capture something of theirs).
I always ask if they want another game after I walk them through area scoring with me. I often get the chance to find the context to talk about two eyes (usually when they try to save a group without nearly enough space), and I'm surprised how many fast learners I get to point out little ko fights with while playing.
I try to avoid jargon unless it's to break a long silence, in which case they're usually curious instead of overwhelmed. As we play more games, I emphasize that it takes a long time to really get a feel for the strategy. Near the end of teaching/playing (or when someone walks up who's heard of Go and/or wants to watch), I mention that there's an awesome wiki about Go called Sensei's Library that they should read!
Dieter: Yes. You can read Dieter Verhofstadt/TeachingExperiences to get my opinion. I basically disagree that one should teach beginners that the game objective is to surround territory, because territory is an acquired concept that is too difficult for beginners. I know that is how it has been done in the West (I also accept that it may be the way in the East, but I believe they have a deeper context for it) but I believe it is a mistake. On one hand it confuses beginners how their territory can be taken away so easily by an experienced player and on the other it poisons them with the disease of thinness. Usually, the territory way talks about efficiency while that is really an advanced concept: "don't put stones in your own territory" (but do so when there is danger).
I'm a strong advocate of the stone counting teaching method and on small boards (I mean 5x5) at that to start with. But, as with everything, use the method you like best! Good luck.
MrMormon: I agree that there may be better ways to teach Go, but I dislike the idea of skirting around (or lying about) the true game when that is what a student actually wants to learn.