Dominion is an incredibly complex game, and with millions of possible board combinations, there is no one "best strategy" that will work in every game. However, these basic strategy tips about money management in Dominion will help if you are just learning to play.
1) Buy Bigger Treasure
For new players, this is the single most important piece of advice you can get. As long as you consistently buy bigger treasure cards, your deck will improve every round, and eventually be able to buy provinces.
In fact, it is possible to win the game by doing nothing except buying bigger treasure and then Provinces. While not very interesting, this strategy (called "Big Money") will actually win against most beginning players.
It is not, in spite of what it may seem like, the best strategy. Far from it; most advanced strategies will easily beat a deck buying nothing but money. But many new players make the mistake of neglecting the importance of Silver and Gold.
Aside from the victory points, Silver and Gold are often the most important cards on the table. And unlike specific kingdom cards, which may or may not be in any given layout, Gold and Silver are always available.
If you want to understand why bigger treasure is important, it can be explained this way: Provinces cost $8. So in order to buy a Province, you need to have $8 worth of buying power in your hand. But your hand is only five cards.
Sometimes, not every card in your hand will add to your buying power. And even if all five of them were Copper, instead of Estates, that's only $5 worth of buying power.
So in order to buy Provinces, it's very helpful to have Gold. Gold gives you $3 of buying power in a single card. If you have two Gold in your hand, that's $6, which means you only need $2 from your other three cards combined in order to afford a Province.
However, Gold itself costs $6 to purchase, which is also more than the $5 you get from a full hand of Copper. For this reason, purchasing Silver is also useful, because it lets you draw hands good enough to buy Gold.
Once your deck is full of Gold and Silver, you'll notice that you have a lot more options of what to buy each turn.
2) Don't Buy Copper
There's a natural tendency to want to buy Copper, because it doesn't cost you anything and lets you add money to your deck. After reading the first tip, you might think that buying more Copper would be a reasonable step to buying more Silver and Gold.
Unfortunately, you would be wrong. While there are exceptions with certain cards in play, generally speaking you never want to buy a Copper. It is almost always better to pass, buy nothing, and skip your buy phase, than add a Copper to your deck.
The reason for this is simple math. You are trying to maximize the value of the average five-card draw from your deck. Every Copper you add to your deck brings that average value closer to $5.
However, as we mentioned in the first tip, an hand of $5 isn't good enough to buy a Province, or even a Gold! To get the really good cards, you'll need an average hand value of at least $6, and eventually $8.
If you buy many Copper, you'll draw hands full of Copper, and that's just not good enough after the first few turns.
The last thing you want to do is water down your deck so it only gives you $5 a turn at best. Conversely, every silver you buy brings your average hand closer to $10, which is a whole lot better.
3) Switch from Treasure to VP
While Victory Point cards are dead weight in your hand, you do need them to win the game. So when should you start buying them? In the case of Provinces, probably as soon as you can. Many players like to have at least one Gold before they start buying Provinces, but once you've bought a Gold, any time you can afford a Province, you should buy one!
As a general rule, once half of the Provinces are gone, it's time to stop worrying about making your deck better with more treasure, and time to start buying Duchies or even Estates rather than treasure cards.
After all, the point of building your treasure in the first place was to get the VP!