I don't recall anything in the existing manuals that allows a "simple" way to detect curses. I believe they were trying to make curses more interesting to play. So, from the rules view there is no easy way to know if an item is cursed except by trial an error (It adds drama!).
An easy way out would be to house rule a curse-detecting clause to a spell (identify, for example), in the sense that it would detect the negative effect of using the item.
Other ways are to hint it to player. Lore, like you said, is a good approach. Legends, rumors, hystorical records could point to a certain object being cursed. Maybe there's a warning in the item's resting place or the PC sense something "not right" when they touch it.
The fact is that by allowing to easily detect a curse it cuts off a bit the point of it being cursed. Curses are meant to be narrative devices. If you want to avoid the hassle you can always reduce the "curse" to negative traits or flaw like the DMG suggest in the artifact section.
I'm going to take a markedly-different approach from many of the others; hopefully this perspective is useful, too.
Start by looking at the nearest-neighbor spell to augury: aura of life. 30' radius, resistance to necrotic damage, regains 1hp at start of 0hp turn. Very easy, and the only thing really left up to interpretation is "nonhostile."
Most spells are like this: they're very clearly... spelled out, using defined terms. [/rimshot]
Augury is a different type of spell.
Augury is a super-vague spell, with lots of room for GM interpretation, preference, style, &c. And you're a first-time GM, so you feel like you've got little guidance from the spell. You're right, by the way.
There's a reason for that: while most spells are a manifestation of the player character's will, this spell is designed to do something fundamentally different. Augury allows the player character to engage in conversation with the GM. Not the player; the player character.
I want to be clear on this: the purpose of augury (and some other divinations) is markedly different than most other spells. Once you realize that, things fall into place.
At some tables it's perfectly appropriate for a GM to say to the players "hey guys, if you take on this dragon you're all probably going to die." If necessary, this is justified in-fiction as "your characters are noting the number of dried skeletons of previous adventurers, they know stories of Smaug from their childhood, they've trekked through miles of desolate wasteland just to get here." Or it's hand-waved away.
At other tables that would be completely unacceptable. It'd ruin immersion, it'd destroy people's roleplay, it'd cheapen the game. Crossing the boundary between in-world and at-table knowledge/conversation/interaction is forbidden.
Augury explicitly crosses that boundary, in a way supported in the fiction. Augury allows the character to talk (specify course of action taken very soon) and the GM to respond: "that should go well," "uh, I wouldn't do that," "kinda mixed bag," or "meh."
But what about timing, subject, and adjudicator?
My best advice is never to spend more than 30 seconds on an augury. Listen to the question/proposition, lean back, think for five seconds, and give an answer. If your players or PCs want better answers, they can wait (or pay) for divination, commune, or contact other plane.
Best Answer
Sure, this seems like an entirely uncontroversial use of the spell.
But do be aware that a GM might have a deity value magic items differently than a PC would. You might get "woe", not because the item is cursed, but because it is considered sacrilegious from the deity's perspective. Or a holy item might get a "weal" result, despite a curse on it, if the deity thinks you're going to be able to purify the item, or if it thinks not having you attune it would be worse (e.g. somebody else would be more vulnerable).
There is also a mild opportunity cost to routinely casting Augury for questions like this. If you have some other event come up on the same day that you might want some divine guidance on, you may not get a truthful answer if you cast Augury again.