This is a very interesting topic, sometimes I have been thinking about it already. Usually I keep hidden whatever I can, just in case

but really it would be good to try out being more open.
Inconsequenti-AL said:
Do they know skill check DCs before they decide to use the skill?
This is something we discussed for a while long ago. In a way it would make very sense for someone who is trained in one skill to be able to evaluate how difficult a task is, at least vaguely. Obviously that cannot be applied to every skill: a climber may be easily able to evaluate if a surface is beyond his capabilities, but you definitely cannot say the same about search, listen and spot checks for example.
However we never tried to play that PCs can know the DC beforehand.
When it's up to me to DM I at least suggest to the player (if they ask) whether a task seems trivial, feasible, very challenging, etc... expecially if there's a dire penalty for failure.
One exception that I do (and here I understand that many DMs would never allow this) is that I always tell the player if Take10 would result in "failure with bad consequences". This can be explained in-character as starting the task and halting as soon as you notice that you're making a mistake. IOW, it's like disallowing Take10 in this circumstance, because it will be fatal. This just because I think Take10 is meant as an advantage given to the PC, and should not become a trick.
Inconsequenti-AL said:
Do they get to know the save DCs they're rolling against?
It's possible that I have slipped out the save DC sometimes, but it's probably not good if they are allowed a way to modify their ST after knowing the DC (feats, reroll-abilities, ...).
Inconsequenti-AL said:
Do they get to know which spell an NPC threw at them?
Generally not. It's more fun to describe the effects and the visual manifestations (which I sometimes scramble as will Spell Thematics) that to just name the spell. But eventually when the DM is tired, there's usually no problem in doing that, unless it's an illusion obviously, or unless someone is trying to counterspell.
Inconsequenti-AL said:
Do you tell them the AC of a combat target? Or get them to roll and you tell them whether you hit or not? Something else?
It's very easy to figure it out after a few rolls, but best not to tell beforehand so at least they have a couple of rounds of mystery

Also often there are abilities such as dodge or selective protections which make the AC different depending on who's the attacker.
Inconsequenti-AL said:
Do you tell them the total dice rolls or just whether the NPC failed or suceeded at whatever they were trying to do?
Normally just the resulting effect.
Inconsequenti-AL said:
I'm seriously considering moving towards a very 'mechanics in the open' style of GMing. I think it'd be interesting to see how this worked out... would certainly give the players a very concise view of the world around them? On the other hand, would seeing 'under the hood' kill off the mystery of the game? Would it make things too wargamey?
Well you can try and see what happens

I don't think the game would be spoiled at all, it will simply become more strategic and less random once you know better which action may be moot and which other may really help. Combat will definitely be less mysterious and more wargamey, but you can keep up the mystery with the story and will many other things (I mean, if all mystery in someone's game depends on not knowing numbers in combat, that's a pretty worrying situation...).