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
Yes, but...
Yes, the Gods can lie to their clerics. In fact, there are spells (e.g. Contact other Plane) that can force a deity to lie to you, with the nature of the lie shaped by the personality and goals of the deity in question. This can lead to circumstances where the deity really wants to tell the truth but has to lie and so might do something like tell the exact opposite of the truth and give indication that the statement is a lie, or contact the Cleric after the spell to correct the misunderstanding, or whatever else as befits the deity's M.O.
However, lying when not compelled to is sort of against the code of conduct for a lot of the sorts of deities that a PC Cleric is likely to follow. I don't know if you are using the default deities or have created your own, but, as an example, if you have Heironious or St. Cuthbert lie to a high-level Cleric of theirs out of convenience that would be really out of character. You should consider having the deity just explain what's up to the character instead. "Yes, I know the answer, but I want you to stay here and not figure it out, so if you could go do that, that'd be great, k thx bye". Or whatever it is that the deity wants the Cleric to do. Clerics are on the same team as their deity, often, and so the deity lying to them would in many cases be really weird and also disrespectful. You shouldn't have the God lie in those cases just because it makes things easier for you; you should figure out another solution to your problem.
Furthermore, all of this has nothing to do with the reliability of a player's Augury or Divination spells, which have their own failure cases (each involves a d100 roll) and do not allow for the deity to twist the result of a caster's answer just on a whim. A deity could deceive a player using such spells regardless, perhaps using Modify Memory or the Delude and Conceal Epic Spellcasting Seeds, but doing so would require active intervention and would be equally possible for any other character with the relevant abilities.