Apprentice is one of those cards that is straight-up powerful. It is a heavy game-accelerator, likely one of the fastest, and has excellent utility overall.
First of all, Apprentice is a trash-for-benefit. All the trash-for-benefit cards speed up the game, but Apprentice does it on a new level. This is because Apprentice
1. Keeps the deck small
2. Is non-terminal, so multiple Apprentices are chainable
3. Cycles through the deck very quickly, both by drawing cards and trashing cards.
Generally, the only reason you should be Apprenticing Coppers is if you already have enough money for what you plan to buy this turn. It's not worth Cutpursing yourself for absolutely no benefit. Apprentice is much better used on cards that cost >$0. Even in the worst case scenario, trashing a 2 cost gives you net +0 cards, which isn't a terrible price for getting rid of an Estate.
As an example, consider Apprentice-trash Silver, which is usually your choice of trashed card. This gives a net +1 Card, on par with Lab. But more importantly, it does so while keeping the deck small. Although you do lose a Silver, there is a good chance of you gaining a Gold in it's place thanks to the cards you just drew. Do this enough times, and eventually you will have enough Gold to be able to afford to trash those. At that point, Apprentice just becomes insane.
http://dominion.isotropic.org/gamelog/201111/19/game-20111119-224955-0e112a2a.htmlThe above is a solitaire Big-Money Apprentice game I played, which shares some similarities with Annotated Game #2. It gets 4 Provinces by turn 12, then does Apprentice -trash Gold/Province before the deck stalls too much to quickly empty the Province the pile. You want more than 1 Apprentice in your deck. I follow "Buy the 2nd Apprentice after the first Gold", although I personally don't know when you should buy it.
The other less explored part of Apprentice is using it to minimize the effect of terminal collisions. Apprentice is unique in that it handles terminal collisions pretty well: just trash one of them! This allows you to overbuy a terminal, play it more often than normal, then use it as Apprentice fuel when you don't need it anymore. See the Apprentice/Sea Hag topic for a potential application. There are likely other cards you could use for this, but whatever terminal you use, it has to be strong, strong enough to make it better than going for Big Money.
Apprentice tends to work the best when you have a source of +Buy to use your large card draws on, which has the added effect of fueling your Apprentices even more. They tend not to work as well in Curse games, where both your supply of >$0 cost cards and your general card quality make Apprentice chains much more difficult.
And just for fun, here's a game of debatable merit where I tried some Apprentice/Trader shenanigans. Lots of free Silver, right?
http://councilroom.com/game?game_id=game-20111103-011729-ff0462cb.html
