Wow, awesome response, and a ton of really insightful information here... thanks to everyone.
I'll try to post my thought process here -- I realize this is basic and trivial to all of you guys, but maybe some other noob will read it and it will help them understand some concepts.
Recently, after 8 or so games of Dominion, I came to the realization that the 10 starting cards are BAD cards, and the faster you can get rid of them, the better off you'll be. I think the extreme case of this is Chapel, which gets rid of these cards super-fast and is recognized as a game-changer.
After reading what you've said, though, I've realized that this concept, which you call trash-for-benefit, is not a "digital" concept (Chapel vs. Non-Chapel), but rather more of an "analog" concept, where each trash-for-benefit card has a certain speed at which it gets rid of the starting cards in your deck. Of course I'm sure that the interactions between this card and others are far more important than this concept, and the side benefits of the card are also important.
All of these things equal, though, it seems that these trash-for-benefit cards have a way to be properly used. Playing successfully with them and watching how they work is an important part of this. I think that's why the content you all have posted is so helpful.
Now that I'm more familiar with this concept, I think that familiarizing myself with the other trash-for-benefit cards would vastly improve my game. It's clear to me that these cards are next to useless when used improperly. Also, the presence of these cards seems to be one of the more significant factors that shapes my overall strategy for each particular game.
It might also be useful to have a general idea of how fast a trash-for-benefit card works, compared to the other trash-for-benefit cards; or possibly this would be useful if we were comparing card combinations instead of just single cards. This might help to gain a good idea of effective ways to trash my starting cards other than just using Chapel.
I think I'll state my understanding for each of the other trash-for-benefit cards in the base set (as a starting point)...
(Chapel) -- Has been discussed at length on here, and has a few awesome articles written about it. I'm not going to embarass myself trying to paraphrase them.
(Remodel) -- We've talked about Remodel. I'll have to play a few games trying this out...
Feast -- This only seems to be good in the early game when there are no other good 4-cost actions available, I have $4 to spend, and I really want a 5-cost card. I'm punished for not having the $5 to spend now by having to spend an action later to get my 5-cost card, and getting to play my awesome 5-cost card one less time than otherwise. If this punishment is less than the relative benifit of having the 5-cost card I want over any of the 4-cost cards, then I buy Feast.
Mine -- This card takes my crappy treasure and turns it into good treasure. It's elegant in the way that it works, in that I immediately get to use my shiny new treasure card, but it's very slow and has no effect on Estates. The main issues I have with Mine are: how many Mines should I have? And at what point in the game does the immediate suckiness of this card (terminal action gets you +$1 and nothing else) start to outweigh its long-term benefits (trashes a bad card, gives you a good card)? I would also think that Mine and Remodel would work well together, since Remodel is best used on Estates and Mine is best used on Coppers, but since they are both terminal actions, it points to buying... Villages! Does this actually work in practice, or does it just become way too slow to work well?
Moneylender -- This card certainly looks a lot better now that I realize that Coppers are bad cards. Unfortunately, it hasn't been available in any games since I had this realization... Is the common usage for this card to trash a Copper and buy a Gold? This seems much, much faster than Mine, and it makes me wonder why I would buy Mine when Moneylender is available. Of course, I'm obviously wrong because Mine costs more than Moneylender; I wonder what I'm missing...
None -- What if no trashing cards are available? Does this mean that my best option is just to try and draw as many cards as possible every turn? Does this mean that Smithy/Big Money is the best move? (I'm sure the answer for the base game is different from the expansions) My original thoughts are that with ten crappy cards cumming up your engine, you want to turn over your deck as much as you can so you can play your good cards more, which points to a *shudder* Village-based engine. People talk a lot more about Smithy/Big Money around here, so I'm inclined to think that's the correct answer, but I'm sort of at a loss for why...