I'm new to Fate (having played it once), but I wanted to look into it as an engine for a Star Trek game I want to run, since it's generic and
skinny (though, as I'm learning, it's certainly not rules-light). I'm reading the PDF, and a lot of things don't make sense to me.
Honestly, I love FATE, but that book I think might be a little too heavy a read for first-timers. Its shooting to cover all the ground that experienced and sophisticated users of the system have come up with. Sometimes I think it loses the simple game underneath. If you haven't seen it, try the "Accelerated" edition. I can (re)assure you that it actually
plays a lot lighter than it reads IME. However, it is a kinda different philosophy for D&D-ers.
(p. 69) You can apparently invoke an aspect "on someone else's character sheet," and if you do, you give them a fate point. So, does that mean you can invoke other characters' aspects to get the bonus for yourself? If not, what the hell does that rule mean? You invoke someone's aspect for them, to give them a bonus and give them one of your fate points? Either way, why wouldn't everyone do this all the time, so everyone gets bonuses (and fate points are just shuffled around between players and not truly lost)?
Not to disagree with [MENTION=177]Umbran[/MENTION], but....I will.

A bit.
Yes, you can invoke other PCs aspects for your own benefit, so long as it makes some narrative sense that everybody/theGM buys. If you do, they get the FATE point. (Unless I'm forgetting something, all other invokes will send their Fate Point payments to the "bank".) You can actually invoke
any aspect that's lying around (from the scene, setting, etc.), but the Fate Point goes into the "bank" unless another player "owns" the aspect.
You cannot (usually) invoke someone else's aspects for
their benefit.*
You
can invoke your own aspects to help another character on their roll: "Take a re-roll, Bob, I'm
Good luck for everyone but myself." That still sends the FP to the "bank".
The text (I think) is a little unclear, but most tables/GMs wouldn't let you invoke a third-party aspect (from the scene, zone, situation, etc.) to help another player.
Mostly, this is to preserve the Fate Point economy. That is, Fate Points are little bits of undefined narrative awesome. (To put it in very crude terms). You're supposed to be earning your FP through compels on your aspects (either from the GM or other players). If you haven't earned the awesome, you shouldn't get the benefit. When you invoke your own aspect for someone else (or vice versa), you're still demonstrating how awesome you are (even if its just from hanging around with this other cool guy).
I mention the "bank" (also called the "pool" or "reserve") because its a distinction that is often confusing for new players. See, the GM has a pile of Fate points, but he can only use them to pay for compels. That's the "bank". Occasionally, the GM might have important NPC's that also have (usually only one or two) Fate Points of their own. (There's a few other odd cases, but their all in the GM's hands and the game plays fine without them.)
*Some games in the Supers genre might have "team" aspects that you could possibly invoke to help a teammate out...I'm a little unclear as to how "ownership" of those aspects works, but then again it seems simple and fair enough to say that if we're both Xvengers we could use that to help each other out, thus contributing to the "team" awesome.
(p. 71) If a player proposes a compel for another player, the player who proposed it has to pay a fate point to the GM. So... why the hell would anyone ever do this, if it creates a bad effect and costs a resource? Just to be a dick to a fellow player?
No. If you accept a compel on your character, you get the Fate Point. Always. In fact, I've usually seen it that if you get hurt by a compel on
any aspect, you get a FP. (Sometimes everyone gets an FP when the fur is really flying.) The only thing that changes is where the Fate Point comes from. The GM has an infinite pool of them (that "bank" thing). Other players have to use their own to compel you.
Why would they do this? Maybe you're out of Fate Points and there's a climactic scene coming up. Remember, this game is not the purely additive exercise that D&D is, nor does it have the same sort of antagonistic GM-players relationship that old-school D&D has. Aspects (and the characters they describe) need to be
interesting, and sometimes you need to take the hit to make things interesting. If this happens regularly, you might want to re-think your aspects and look for something that's more easily compellable.
Are there any Fate gurus who can help me figure this stuff out?
I dunno if I'm a guru, but I'll try to help out. Also, there's a thriving Google+ Fate Core Community (like 4k members). Its basically sucked up all the activity from the old mailing lists and Yahoo group.