One thing that helped my family tremendously was to play the first few games with everyone's hands shown all the time. So you can explain as you go how the actual mechanics of playing Dominion - the ABCDs (Actions, Buys, Clean-up, Draw) - and at the same time demonstrate how the various cards actually play out. I also helped explain the pros and cons of each player's choices, especially when they were confronted by multiple action cards to play.
I also took the time out after every game to ask the newest players which cards in the set they understood and felt confident about. Any cards they were uncertain about - either the mechanics of how to play them, or WHY they would ever want to play them (cough Chapel cough) - I kept in the mix for the next game. As to the rest, I would swap out maybe 3 or 4 cards for new ones they'd never seen before, while keeping the remainder (so they wouldn't be COMPLETELY at sea when the next game started up).
A note on Mine: in my experience, new players tend to have trouble grasping trash-for-benefit in general (so Remodel as well). My mother's mode of thinking for the longest time was, "Well, I could get rid of this card, sure, but isn't it better to USE that card instead? Why get rid of a Copper to get a Silver when I can just use the Copper plus this other action card (Smithy/Militia/what have you) to help me buy something better this turn?" It's the typical bias that all starting players have of not wanting to lose their precious, precious cards. So how to get around that? A few things I found helped:
- Set up a game, for instructional purposes, that involves Chapel and rewards going for a Chapel-thinned engine. I find Adventurer works well for this (even if not maybe the strongest choice) because it makes the benefits of dumping Coppers so glaringly obvious. When they ask how your deck is doing what it's doing, pause the game at a reshuffle and deal out your whole deck. Let them see how few (if any) Coppers you have left, and let them realize for themselves how this drastically cranked up Adventurer's effectiveness.
- Moneylender. Moneylender I think is the easiest trash-for-benefit for players to grasp because its benefits are so immediate and obvious: instead of getting $1 from that Copper, you get $3! Incredible! But at the same time, explain/demonstrate to them that the benefit from Moneylender isn't JUST getting $3 instead of $1; it's also getting rid of Coppers that clog up your deck. If they've seen the Chapel/Adventurer example already, they should start to see how it's a slower-but-surer method of getting the same benefit.
- Now move to Mine. By this time, they should start to be more comfortable with understanding that it's better to trash a Copper and pick up a Silver than it is to use that Copper to help buy a Silver. The card's text itself is a bit confusing, but if they understand the idea behind the card, they'll better be able to wrap their heads around its text.
- Finally, Remodel. This card, out of all the other cards in Base, I find induces the most AP in new players. "What card should I trash? What card should I gain instead? Is it better to NOT trash anything, and instead play my other cards?" It's not helped by the fact that Remodel's best targets are NOT Coppers (Estates are OK), especially in Base where there's no $2 card you really want lots of. Remodel is better used on GOOD cards. But a new player will naturally prefer to USE those good cards instead. After all, isn't that why they bought them? To use them!
I floundered a bit teaching Remodel myself; I first tried a turbo-Remodel strategy on one board as an example, but it wound up underperforming (as turbo-Remodel without support tends to do) and only confused my new players further - "if he can't get it to work, how can I?" What worked out better was getting up to Golds in a more "normal" fashion and then picking up Remodels mid-game to turn Golds into Provinces. That is the sort of "obvious" use of Remodel that helps a new player understand why you'd want to lose something GOOD in place of something BETTER.
Finally, as to handicapping: Always have the least experienced person go first. They will complain. They will say, "but I don't understand the cards yet, I need you to show me." Stick to your guns. If you go before them, they will copy you without pausing to think of any sort of strategy, and then wonder why their deck is flailing around. Tell them to look at the board, think about WHICH cards they are going to buy, and then set out to buy JUST those cards. New players need to understand how much having a plan from the get-go will help them during the course of the game, and the best way to do that is to get them to make those plans themselves. Even if it's just "well, I'm going to try to learn cards X Y and Z this game," that's a good start. Then, pursue some other strategy yourself. Explain what you're doing while you're doing it. When the new player is confronted with a difficult choice, remind them of their original strategy and get them to answer what option(s) fit best with that strategy. It takes time and effort, but it really pays off when that new player looks at you after the game and says, "Hey, I actually understood what I was doing that game." That's the goal!