Googling reveals several sites listing standard tactics for Dominion. Let's collect ideas here, and discuss and judge them using the voting system.
One tactic per post please.
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Remove Your Bad Cards
This involves slimming your deck down as small as possible by trashing or otherwise removing your starting Estates and Coppers (as well as any Curses you have ended up with).
Common cards that can enable this strategy (with set in parantheses):
Multiple Card Trash:
Single Card Trash:
Remove Cards From Play:
Removing your starting Copper and Estates will let you draw your more powerful cards that you've bought more often.
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Draw A Lot Of Cards
This involves chaining actions together to draw as much of your deck as possible every turn, letting you use your good cards more often.
Common cards that can enable this strategy:
Any card with +Cards and +Actions on it can be chained with other such cards to keep searching your deck until you find your high-value Treasure. Cards like Laboratory (+2 Cards, +1 Action) are even better.
Other cards that combo into this strategy are things like Scout that will clear out upcoming Victory cards so that everything you draw is playable.
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This involves just buying Silver and Gold, skipping any Action cards in favor of building up your Treasure as quickly as possible. Once you've built up a deck full of Gold, you buy Provinces.
This strategy can be quite successful, especially against beginners, but a good strategy using combos of Action cards can beat it.
Identify the combo
There are two kinds of Dominion decks: Combo decks, and non-combo decks. The line is definitely fuzzy, but in general, it comes down to whether you are aiming to win through a particular set of powerful interactions, or just by having a lot of good cards.
Neither kind of deck is inherently better, but combo is much scarier. If you go into a game with +Actions, +Cards, and Bridge, and you just kinda meander around buying various good cards until you can buy a Province every turn, you're going to suddenly lose from your 3-province lead when your opponent hits critical mass and combos out, making provinces cost $2 and buying the last five at once.
Therefore, it's important at the very beginning of a game to identify any available combos, and decide whether you'll be going for them or whether you think you can beat them by building for pure value.
If you decide to go for a combo, your card valuations have to change -- if you're trying to assemble Torturer/Native Village (and you really, really should be if those cards both show up) you're going to have to pass up that nice-looking $4 for a measly $2 NV a bunch of the time. Caravan is a fabulous card, and maybe your deck should have a couple, but it doesn't help you play three torturers in one turn, and that's where your value is really coming from in this deck. If you already have four torturers and only 2 Native Villages, you need to spend your $5 on another NV.
Some common combos to be on the lookout for:
Bridge -- The mere presence of this card isn't enough, but it's enabled by too many things to list. You need a way to play four-six of them in a turn, which means plenty of card draw, and either plenty of actions or Throne Room/King's Court. It can be hard to get going, but when it does, maybe the most explosive interaction in the game. See this example game from Isotropic.
Torturer and +Actions -- If this is present, it's almost always too powerful to ignore. Forcing your opponent to choose between multiple curses or a 1 or 0 card hand every turn is just disgusting. Example game.
Warehouse/Treasure Map -- Normally a bit too unreliable to be really good, Treasure Map becomes amazing in the presence of fast deck cycling that leaves you with an action. Cellar is also great for this.
Venture and good trashing -- A deck whose only treasure is five ventures and a gold is pretty unstoppable.
Remodel/Gold and Expand/$5s -- Spam lots of remodels, remodel some of them into Golds, then remodel the Golds into Provinces... and then remodel the Provinces into Provinces to end the game once you've gotten ahead and no longer have any buying power.
King's Court/Throne Room chains -- There are openings to a turn scarier than King's Courting a King's Court. The best part of chaining these action duplicators is that you don't need +2 Actions cards to go off -- you can either net Actions by duplication Great Hall, etc, or you can just forgo +Actions altogether: if you can consistently KC+KC+Smithy, you'll draw 9 cards and get to triplicate two of them (and hopefully at least one was another KC...)
Figure out what you're going to do with all those green cards
Having a plan for how to keep your deck quality from degrading drastically after you start buying Provinces can make a huge difference. Just because you draw 8 two times in a row doesn't mean you ever will again, now that you've reshuffled with those provinces, and if you stall out and feel the need to grab duchies to keep up, it just gets worse.
Island, Bishop, Salvager, Remodel, Apprentice, Native Village + top of deck manipulation, Vault, Cellar, Warehouse, Minion, and Venture are all cards that can help with this in different ways.
Figure out if it's possible to get multiple VP sources in one turn
In a spread with no +buys, no +VP chips, and no card gain, (and to a lesser degree, no cursing and no Outpost), it's almost impossible to catch up from behind. In these games, you want to consistently get to 8 in a turn AND NOT MORE and start buying Provinces more or less as soon as possible. Be sure you identify this feature of the spread right away, so you know whether it's safe to try and build up resources to the point where you can get 2 Provinces per turn.
General rule that applies to any specific strategy of play:
Plan your first four turns in advance.
Once you have your opening hand ready, you should plan your first two purchases. Then you have a decent chance of planning turns three and four based on them.
Use these key early turns to mark a clear path to your first Gold and/or powerful 5-cost card.
Watch what other players buy
Particularly attack cards when defense cards are in the game.
Also be aware of how close to running out the piles are. Remember, the game ends when any 3 piles are gone from Supply (non-Supply piles like Cornucopia's Prizes or Dark Ages's Spoils do not affect this), when all the Provinces are gone, or when all the Colonies are gone.
Buy piles other players want
This is related to the above. In particular, don't let one player buy down a single pile, particularly if said pile has plus actions.
Strategic deck building
At the beginning of the game, you need to plan your deck. Find cards that will work together well. But choosing cards isn't all there is to it, it's very important to learn when to purchase certain cards. There is no reason in the world to purchase a Village in your first few turns. Likewise, you'll rarely want to buy a Chapel in the later stages of the game.
In most games, you can break the game down into three phases. Some kingdom cards can change this approach, but in general, this is how I think about the game:
Look at the ten kingdom cards available and learn to distinguish which are key "jump start" cards: cards that help you get to bigger purchases as fast as possible. These cards are key for your first three or four times through your deck.
Since your deck is still small, you will cycle through it quickly in the beginning, and they will come up frequently. You generally only want one or two of these jump start cards, then fill out the rest of your purchases with Silver. Occasionally, a good jump start card will also work as a strong mid-game card (such as Bridge).
Your goal in the early game is to get to your mid game strategy as quickly as possible. Often, but not always, this means getting enough coin in your deck to enable you to purchase cards that cost 5 or more in nearly every turn.
These cards can (and should!) be purchased in your first two turns. These cards generally cost 3 or 4 coin and give you a fast way to afford the out-of-reach 5-cost cards: Moneylender, Smithy, Bridge, Chancellor, etc.
Low-cost Attack cards serve a similar function, by slowing down your opponents before they reach their mid-game.
This is when you build the strategic deck you've planned from the available 10 Kingdom cards. If you played your early game correctly, you should now consistently have the buying power needed to do so. With the exception of terminal actions, you will probably want several of these powerful cards. This means Golds & Platinums (first and foremost), Festivals, Grand Market, Tribute, Peddler, etc.
By the end of mid-game, your deck should be the powerhouse you envisioned at the beginning. You have cards that synergize well and you can purchase at least a Province on most turns. When you have that, pull the trigger on your late game.
Buy victory points! Now that you've build your dream deck, use it for its key purpose: grabbing up Provinces. You can mostly forget about your deck-building here, and just focus on the green cards. On turns where you can't afford a Province, perform deck maintenance: buy another Gold or other mid-game action card so you don't lose too much power after you shuffle in the victory cards you've been buying. As the supply piles dwindle, don't be afraid to cannibalize your deck by Remodeling Golds into Provinces or Forging your powerful cards into Colonies.
A note on trashing cards
Some cards (Chapel especially) let you focus on narrowing down your deck by trashing unwanted cards. Often, your best bet is to start with this immediately, prior to your "early game" strategy. All the cards you want to trash are right there in your deck at the beginning of the game: Coppers and Estates. Before you add too many "good" cards to your deck, get rid of these while you are more likely to draw them along with your Chapel or other trashing card. (This same approach works well with Treasure Map: buy two or three treasure maps right off the bat and then buy nothing until you can cash them in.)
This may seem counter-productive, as you spend several turns without progressing through early- and mid-game, but once you have concentrated your deck, you should be able to get through them to your late-game much faster.
Cards that don't fit these definitions
There are also plenty of cards that don't fit into any of the above categories. Village, Great Hall, Pawn, etc. These don't hurt your deck (since they often have +1 Card & +1 Action), but they do little to help you progress through the phases of the game. Buy these only when you can't afford any truly strategic purchases: when you think your deck is on its way to the next phase but you just got a bad shuffle, and yet another Silver would just dilute the Golds you are starting to buy.
Since I'm not yet allowed to edit even CW posts, I'll just point out that Steward is one of the best options for the "trash your cards" strategy: it's most of a Chapel that can also be a Silver once you don't need to trash any more. A friend of mine who loves Chapel-style decks argues cogently that it's the best card in the whole game so far for this kind of strategy.
Properly manage your reshuffling frequency
An advanced strategy is in certain situations to take cards like shanty town or lookout on turns 1/2 in order to increase your rate of reshuffles, and therefore give more value to each card you gain because you will draw this card sooner than you otherwise would, and you also get to use it more times throughout the course of the entire game.
Another very important thing to consider is during your turn if you run out of cards, to decide whether it's good to reshuffle your discard pile into a new library or not. Sometimes it's can be really good because this will set up your next few turns in a good way, something like all money for two turns and then youll have all your actions from this turn stacked together after the money. Most of the time however it's a lot better to get your library to zero cards and leave it there, that way you don't get any clumps that will hurt you later on.
Check out this post of mine from BGG for some more detail on this point.
Damage your opponents' decks and hands.
Use attacks like the Witch, Spy, Bureaucrat, Ghost Ship, Swindler, Militia, etc. to make sure the other players are always playing with fewer and worse cards than they would like. Especially with the Witch, getting an early start at haranguing the other players can ensure you get a significant lead right off the bat. On the flipside, the Witch is a terrible card once the Curses run out. It's really nice when Upgrade is also in play so you can flip your Witches for Golds once this happens.
Similar to "Remove Your Bad Cards", the small deck approach is not just about bad cards.
It's about keeping the deck size as small as possible to cycle through your deck as soon and often as possible (ideally, getting those trusty golds/actions in play more often).
With the small deck approach, I will typically buy fewer actions, build out a solid set of golds, and then start hammering on the provinces.
Buy a small set of different cards
This does not necessarily mean you need a small deck. What I mean is pick roughly 3-4 cards to buy and only buy those.
If you only buy from a small set of cards then those cards will more frequently pop when you draw and you will have consistent gameplay from one hand to the next. If you have 8 different cards in your deck then you will find each hand will have random cards, which is almost never a good thing given synergies don't exist between every pair of cards.
For example, in one game I only bought Torturer, Festival, and Throne Room. I got consistent play from one hand to the next (Festival for actions, apply Throne Room to Torturer for draw, repeat). To accommodate filling my deck with Provinces I simply bought more of those 3 cards to increase the size of my deck, and thus reduce the number of Provinces I get with each hand while maintaining consistent gameplay.
Get lots of victory cards
Buy lots of victory cards, such as a Duke(I)/Duchy(B) combination, or Gardens(B). Since this heavily dilutes your deck, it only works with action cards that allow you to get cards without having many useful cards, such as Ironworks(I) or Remodel(B) or Workshop(B).