Start off with just one suit. The pin (circle) tiles are ideal because they're the most visually intuitive. Tell the participants that there's four of each tile, that the number tiles go from 1 to 9, that there are in fact more suits but we're not going to bother with them right now. Don't even bother with building a wall, dead wall, dora indicators, points or anything of the sort: get everyone to put the pin tiles face down and shuffle them. Throw all the rules and elements of Mahjong (calls, yaku, etc) out the window, as you'll be introducing them gradually.
Teach them about sets and the pair and how the set can be three consecutive numbers or three of the same number.
Everybody picks up four tiles: the winning condition here will be just one set and one pair. Randomly choose someone to start picking up, building the hand and discarding accordingly, and teach them how turns always go counterclockwise. Make them play with their hands face-up at first, answering questions as they so need. Win by self-draw only, hands usually last less than a minute each. Every time someone wins, ask them (after congratulating them!) how they reached that hand, what they were waiting for to complete the hand and if that was the only tile they needed to win. That's how you start introducing them to the concept of waits, rearranging a hand into multiple possible configurations, and tenpai.
Once they feel comfortable, let them play with their hands hidden to the rest (they'll go crazy when they find out you can make the tiles stand on end). If they can figure out when they're one tile away from winning, you can teach them about ron, how it has to be immediately after someone's discard (this will keep them on their toes) and force them to start using the terms ron and tsumo. You can now begin to make the game a bit more tidy: teach them how to keep a proper discard pile, force them to pay attention when their turn is, and so on.
After several hands like this, make them play with seven tiles - now they have to complete two sets and a pair. Because there are now more ways to rearrange your hand it'll perhaps be convenient for them to play a few hands showing tiles; in this case, forbid winning by ron for the time being as what's now important is to get them used to recognize winning tiles, waits, different hand arrangements, etc. Once that's out of the way they'll probably be asking you to play with their tiles hidden to each other; in this case, reenable winning by ron.
Now add the sou (bamboo tiles). This in fact makes things easier when it comes to hand building. Teach them that the sets and pair have to be made out of tiles of the same suit but the hand doesn't necessarily have to be just one suit. Show them the 1-sou which will almost inevitably look kind of funny in your set, and the 8-sou which is not as self-evident as the rest of the tiles. Make them play with seven tiles still.
Upgrade to ten tiles once you think they're ready. Get them used to ten tiles.
Upgrade to thirteen tiles. Eventually, introduce them to calling pon; while we're at it, teach them proper calling ettiquete. You may want to delay teaching calling as much as possible until your students get the hang of building hands on their own and working on their own intuitions about tile efficiency, but if time is of the essence and you feel you want to get them engaged with wild calling, teach them about pon.
At this point you can add the dragons. Show them how they're their own special kind of suit and because they don't have numbers they're only good for triplets or for a pair. This adds a few more tiles and the possibility of ryuukyoku becomes more tangible.
The rules of Tibetan Mahjong end around here. From here on, all there's left to add is aspects of the game proper, such as:
Man tiles: The extra suit won't be a problem, but here comes the hard part of memorizing number characters. I am generally against adding arabic numerals to the tiles because it's a crutch people have a hard time getting rid of, but if you deem it necessary and you trust that your players will do the memorizing homework, then it may even be beneficial.
Yaku: Start with whatever's available with the tiles you have in the order your players will feel more comfortable with - toitoi, tanyao, dragon yakuhai, chinitsu, honitsu, ittsuu, etc. Make sure to tell them now that a hand also needs to at least have one yaku to win, but don't force them just yet to do so, merely encourage them. A long list of yaku can be daunting very quickly. Remind them that one hand can have several yaku depending on composition. Don't even dare to mention pinfu.
Wind tiles: I'd only add these well after they're comfortable with yaku and if they're willing to memorize the kanji, though letters could be added to the tiles to help (with the same criteria as doing so for the man tiles, mentioned above). Teaching them about seat wind and prevailing wind will be confusing at first and definitely not something that can be easily digested on the first day.
Scoring: Keep it simple and count the fan only, starting at 1000 points with 1 han and doubling until capping at mangan. - 1000 → 2000 → 4000 → Mangan. No kuisagari fan deductions (i.e., don't make a hand be worth one less fan if it's open). Remind them that scoring is zero-sum and points are passed around, and that a player who deals into someone else must pay the total sum of the hand. This is where the bloodshed slowly begins. Tell them that dealers get a 50% bonus and that scoring is in fact A Rather Convoluted Affair With A Fair Bit Of Math Going On But Nothing That Should Concern You Guys For The Time Being Besides Nobody Remembers That Damn Scoring Formula So People Just Memorize The Score Chart By Inertia. With teaching scoring also comes teaching about ryuukyoku payments; this will reinforce the necessity to keep tenpai and your players, knowing they can deal into someone else's hand, will delicately taste the forbidden fruit of mahjong recklessness: “Should I fold and break my tenpai, paying at most 3000 points - or should I keep going to see if I can win first, risking to pay more than twice this amount?”
Furiten: By now everyone should know to keep their discards separate and tidy (otherwise calling tiles would be a messy affair). Furiten can be a bit of a disaster to explain from the get go (people will inevitably fumble with waits a little bit) so add it gently: forbid a player from winning by ron with a tile they have previously discarded, but don't forbid winning with other possible waits; e.g.: a 258p wait with a discarded 8p forbids winning by ron with the 8p but not with 2p or 5p. Because furiten is the foundation of defense, underline the players the fact that not only they can't ron on a tile they discarded - their opponents can't either. This will give them (by intuition or by explicit indication from you) the concept of betaori, a basic tool for folding: if you don't want to fall into your opponent's hand, discard tiles that your opponent has previously discarded.
Chii: It has to happen, eventually. As with the others, make sure you double down on the enunciation of calls: make them all very vocal, very quick, and have them point if they need to.
Wall, Dead wall, Dora, Kan: Adding dora to the game implies adding a dead wall for the indicator. Adding a dead wall might as well imply setting up a live wall, and having a dead wall and dora will invoke the necessity of teaching kan, new dora indicators, and so on. Because it all comes into the same package, it's gonna get a little messy.