Never. (Note: Many expansions have been released since I first wrote this answer, there are more exceptions than I list here, but the general principles apply.)
Well, almost never. If you have to ask, the answer is "never". Don't buy copper unless you've got a specific strategy that will make it worthwhile. Some new players have a strong urge to use all their buys, or at least to buy something every turn, but buying nothing is better than buying Copper almost always.
OP mentions the Gardens exception in the question; other cards that can be used effectively with Copper include Goons and Forager. If your target buys aren't too expensive (say $5 or less), then Coppers can help you hit $5 consistently; this applies to Duke/Duchy, Gardens, Ill-Gotten Gains, etc. If, however, you are hoping to buy cards costing $6 or more, Coppers hurt.
New players sometimes make the mistake of thinking more money is better, so they'll buy a Copper with a leftover buy. This is bad. The money in your typical hand is what matters, not the money in your deck. With 5 cards in your hand, adding Coppers is going to
reduce your buying power unless your deck is really bloated (maybe with Curses/Ruins). New players quickly learn to hate a hand full of Coppers when they want to buy a Province and stop making this mistake.
Just think, when you draw a card, are you thinking please, please, let this be a Copper? If so, then maybe you should be buying Copper. If you'd prefer a Copper to what you typically do draw, go ahead an buy Copper. More often, you are hoping for something better than a Copper and you are disappointed if you draw a Copper - this is your cue to not buy Copper.
There are cards that make Coppers better, but players with some experience often overestimate their power. Examples include Moneylender, Coppersmith, Bishop, Mine.
Moneylender can be a great card. Some players will think it's a great card because of the $3, but the real value of Moneylender is the trashing of Copper. It shines in the absence of more efficient trashers because it cleans your deck of Copper. Since Moneylender is a terminal action, buying several Moneylenders and then buying Coppers to feed them is not a viable strategy by itself. It requires 2 cards and an action to generate $3; a hand with two Silvers instead of Money Lender + Copper has an extra $1 and an extra action. Even Silver + Money Lender without a Copper can be better than Copper + Money Lender if you've got a different action card worth playing. Spice Merchant and Miser are similar.
Coppersmith - there are decks in which Coppersmith can work very well (@BlairHippo brings up Apothecary/Coppersmith), but it takes more than just Coppersmith. You need a lot of card draw to make an engine out of Coppersmith, but card draw helps almost everything so there are usually better strategies on the board.
Bishop can convert Copper to VP, which is great. But anything costing $2 or more gets you more VP. With extra buys, you'd be better off buying Estates than buying Coppers to feed to a Bishop.
Mine is like Money Lender in that what makes Mine a good action is that it gets rid of bad treasure, replacing it better treasure. If you buy more bad treasure to feed to a Mine, you're missing the point.
The themes above are the some cards let you trash Copper for some bonus. The main value in using them is trashing Copper. If you buy more Copper, you are undoing this work.
- Counting House/Travelling Fair is an effective strategy that is Copper-centric.
Of course, if you have a bunch of activated Cities (lots of actions and cards) and a few Goons (lots of buys, rewards for buying), go ahead and buy some Coppers, and throw in some Moneylenders/Coppersmiths/Spice Merchants too! But if you've got a good Goon engine going, you don't need me to tell you that you shouldn't normally buy Copper.