In the case of some of those checks, they're kind of automatic (or passive) -- so, yes, I would prompt the person with the appropriate skill to make a roll -- and, in fact, doing this a few times might help them realize those skills are useful and prompt them to try it themselves later.
In the real world, someone talking about a topic will make you think about what you know of that topic, in order to integrate the information -- so, in game terms, when getting part of a story, people would be making automatic/passive knowledge checks about what they're hearing -- and, if successful, assuming they're listening, the stuff they know about that topic will kick in and result in them questioning this "new" information if it doesn't jive with what they know.
(A lot of skills would work this way -- perception, for instance, I mean, you're always looking around and listening, even if you're not doing so intently.)
In addition to those gentle reminders, though, it might be worth exposing them to an allied NPC sage/expert who actively makes use of those skills -- it's a not-too-subtle reminder that they can do it too -- seeing the "expert" appraise their gems, negotiate for more money (to prove his worth to them), or -- better still -- identify the creature and a weakness that it might have -- will likely leave some impression and make them more likely to ask questions and hopefully reach for their dice.
Absolutely. Most GMs I play with usually ask for those rolls if it doesn't occur to the player to ask for them --- also, there's a matter of determining what's common knowledge/an automatically assumed ability and what requires a check. For example, I might say to the GM, "what do I know about that king you mentioned who beheaded his second wife?"
He might ask for a Knowledge History roll --- or he might just answer the question, on the rationale that it's such common knowledge it's not worth checking. This might be important indeed if it's information the GM really wants the group to have. But I feel it's my role as player to leave that decision to the GM, not assume I have to roll.
Indeed, a lot of times players assume no roll is needed unless the GM explicitly asks for one. It's not like you need to roll a check for tying your shoelaces, so they might be waiting to hear from you what kind of circumstances warrant bothering to roll.
Sometimes you can make skill checks for them. Things like Sense Motive can be done without you telling them that you are making the check in the first place.
I let the players tell me what they are going to do then I tell them what type of check they need to make. So if they are just trying to convince someone by being nice, then it's Diplomacy. If they are trying to force someone to do something, then Intimidate. If they are trying to lie, then Bluff. I usually let my players make their own rolls because they don't metagame. However, you should make these rolls behind the screen so they don't have an opportunity to metagame.
Skill checks are another that you can do without telling them that you are doing it. For the most part, I assume that the characters are Taking 10 when not stressed. That means that if they have ranks in Knowledge (history) I can tell them what I want them to know based on that Take 10 check. If they want to know more, they can roll.
I've also been known to hand out different bits of information to each player based on their character's skills. This is something that requires more work but it is really interesting to watch the players realize that their characters know different things and then they start to look at their sheets to see why. This is really fun with NPCs. A few different notes on NPCs based on Take 10 Sense Motive checks done before the game for NPCs they have known for a while is interesting.
You will want to have a sheet with all the characters skills handy. You can find a useful one here. It's for 3E but it should work fine. You may even want to just make one that suits your needs as GM.
If something is important, IE would lead them to a major plot point, absolutely tip them off. If not directly remember the rule of 3. If something is important you should give the players 3 different routes to get to it/come up with it. Never leave things to a player remembering to make (and in fact succeeding on) a specific check.
Here's the thing, I'm going to be playing in a PF campaign soon and am eager to start my own. The problem is that if the players don't see a straight line to a goal, they'll never think to find their way to it (nor would they try and find flavor on their own). For example:
If someone went on and on about how the region has gone through such and such, they'd never consider a knowledge history check (even if a racial made such a check a class skill and trained in it, such as one of the dwarven racial archetypes) to see what they can find out, they'd just take the info straight from the guy talking to them.
They find some loot and just accept whatever value a treasure generator states it is worth or whatever the guy they want to sell it to says, no consideration to an appraise check.
If I put in an NPC, they'd most likely assume he's an ally, quest giver, enemy, objective (of a quest) or completely unimportant (pretty much they'd sit there thinking 'is there a point to him talking to us?'). They'd listen to see if the NPC complains of something they can deal with, if they don't get that, then they assume the NPC is a potential enemy and sense motive/bluff/perception him like crazy.
These are guys who'd probably only ever use perception, sense motive, diplomacy, bluff and maybe intimidate. They have no initiative to use any of the other skills and I was wondering if maybe I should prompt/tip them to make use of some of these skills to make more of the lore and flavor come to them. I was thinking something along the lines of 'you think you may know more about this' (and if that doesn't help, flat out tell the guy to roll a check), or 'something catches your eye' (then they'd ALL reach for their d20's).
One way to deal with Knowledge checks is for the GM to know the Knowledge areas (with modifiers) that all PCs have ranks in. Then when a Knowledge check is required, you ask everyone (or specific PCs) to roll a d20. You then calculate if an appropriate Knowledge check was successfully made.
Then you tell the successful player that their PC just happened to remember (typically based on prior experience, knowledge, something they overheard, read somewhere etc) the appropriate Knowledge detail in question.
I think this lets you weave the Knowledge check into the overall narrative better than reducing the check to DCs, die rolls and modifiers.
This works especially well for monster identification checks. Asking for a Knowledge (Dungeoneering) check gives away the fact the monster is either an aberration or an ooze. But you can conceal that bit of info if required as part of your reveal if you don't identify the specific Knowledge area that triggered the reveal.