I have been trying to get several people to play Dominion lately and have taken the following tact when explaining it.
First, I only lay out victory cards and money (with no curses, trash pile or actions) arranged by cost in a row in front of the player. Then I explain that the goal is to get the most victory cards and that they must use money to buy. I explain that this game is like war and that you are trying to get the best deck. I then explain that the game ends when all the provinces are gone (I mention the 3 pile rule later). I also explain how Dominion is different from normal games in which you must "turn in" your money in order to use it. I play 5 turns (that way you get a partial deck reshuffle at the end of turn 4) for myself with this setup to explain buying (you can't use it now, it's for later) and reshuffles and drawing and only buying one thing.
Then we begin playing and after about 4 turns, I introduce Smithy as an action card. I re-explain that this card is for later and encourage them to just buy money if they feel confused. Then, 1 by 1, I add Market, Cellar, Adventurer and Woodcutter (they are all fairly straightforward, each cost a different amount). After playing a full game which is basically big money + a couple actions, I ask if they would like to play again with the same actions or a full set of 10. This is when I bring up the 3-pile victory condition.
I will generally then play 3-5 games with just the base set and the following time playing together add Hinterlands cards (Sometimes I jump ahead to this step with people who grasp the game easily).
Introducing the game this way generally teaches solid fundamentals and players are less likely to make mistakes like drawing from their discard pile or reshuffling at the wrong time if they are taught in this way.
I find that fronting the Big Money strategy does 2 things:
1) It skips the Village Idiot phase that every new player is generally inclined to go through.
2) It makes the player competitive out of the gate, which means that they are having more fun than losing and more likely to figure stuff out for themselves.
I like to add the sets in this order:
1) Base Dominion
2) Hinterlands
3) Seaside
4) Prosperity
5) Haven't gotten this far with anyone yet.
So far, it's worked extremely well. I've gotten to teach even non-gamers and they love it. This weekend, I got together with two people who own Dominion and a couple expansions and a new player. I walked through this with the new player (without actually playing the first game with them, just explaining). We played 3 games, 2 base Dominion games and 1 Dominion/Hinterlands split. The new player won 2 of the 3 games, and I won the other.
I found this method online somewhere and it has not failed to get people into Dominion yet.
Hope this helps
. They weren't even so dominant that each player NEEDED them, and they were quite fun and especially useful in the last game I posted above. Ultimately, I found them to be mostly situational (Sage seemed the least so), like almost every card in Dominion.