There are at least 4 ways to use FATE aspects:
- Tag opponent for bonus to self - you pay 1 Fate
- get tagged by opponent for penalty to self - You get 1 fate
- get compelled by opponent to force a move or prevent a move or attack- you get 1 fate if you accept, pay 1 if not
- get tagged by opponent to who narrates the outcome - You get 1 fate if you accept.
To illustrate these, I'll borrow a scene from Return of the Jedi.
Lando is fighting one of Jabba's henchmen, with Han and Chewie right nearby.
The henchman has just hit Lando.
The henchman wins a maneuver to cause Lando to be Off-Balance. Lando now has that tag.
The GM, realizing that this is just too good to pass up, tags that to lower Lando's initiative below the henchman. Lando's player accepts the fate point. The GM then decides to have some fun, and shoots the skiff. He calls for a roll on athletics to stay upright.
One of the bystanders, however, has a bright idea, and gets the GM's permission to offer a fate chip... to Lando to fall off (a compelled move). Lando's player accepts.
Han's player also blows the roll, and goes prone.
In fairness, the GM rolls for the henchman, who also fails, and decides he's not interesting anymore, so off he goes, too. Han blows a fate, and offers, "Not just down to shoot at Lando, but he fell into the sarlac... munch! munch!" The GM accepts this.
Han decides to grab Lando, but fails the roll. The GM ascribes it to Lando being grabbed by the Sarlac. (Fate point to Lando for narrating a truth about him.)
Han decides to shoot the tentacle, a maneuver to get a free tag on the sarlac, with which to compel a release, but still has that blinded injury. (Injuries can be tagged for free...) Lando helps (by describing Han's aim), and Chewie helps by preventing Han from falling. The GM offers Han's Player a fate to compell him to shoot Lando instead. Han's player thinks for a moment, and says no... So the GM uses the free tag on injuries, and han, of course, misses.
Han tries again the next round, and spends a fate activating his Blaster Master aspect to counter that injury penalty and tries again, and makes it. The Sarlac is labeled, gets compelled by lando, and it lets go. Han makes an athletics to pull him up.
As you can see, it's all about how you offer the FATE. If you offer a compel, it's a gamble... but it's good story. If you tag for a bonus, it's not - pay them, take the bonus, and go.
The one caveat, as I was reminded by Seven Sided Die, is the Authority, be it GM or Table. If a compel or tag is nonsensical, inappropriate, or simply bogs the story, it can be rejected by the Authority. In some FATE games, this authority is the GM's; in others, the authority is the group, and explicitly not the GM's alone. In either case, if an aspect is tagged for bonus, or for penalty, barring "That makes no sense" by the authority, it's a done deal, the fate moves. Compels for specific actions or against specific actions are always subject to review by the authority, and thus have 3 outcomes: Rejected by authority (no fate moves), accepted (Fate to compelled player), or rejected by recipient (Compelled player pays proposer of compel).
Noting that the compel portion can be an absolute bar - for example, blind. Many things, blind is a penalty to. But, for example, to read a book, just compel it with "can't feel the letters, can't see the letters, can't read it."
It's also worth noting that a player rejecting a fate chip for an obviously legit compel probably should be whacked with derision and peer pressure by the rest of the group as it's a clear break from the fiction.
When one is invested in failure, one takes the position that the goals of the player and the goals of the character are not always the same. Your character wants to succeed at everything they try, while you as a player want them to fail now and again -- not just because it makes things more interesting, but because doing so earns you Fate Points to guarantee success at other, possibly more important things.
In this way, accepting compels and making concessions are "investments"; you control the circumstances under which bad things happen--although not the details of how--and they pay dividends later.
However, the other payoff from failure is, as @SevenSidedDie observes, that the story does become more intricate, more involved, and more engaging. (As a Fate fan, I would also say "more fun," but that's subjective.)
Best Answer
Fate has a much more narrative approach, less GM authority, and player-based plot control mechanisms. This can make for trouble transitioning from a more adversarial GMing environment.
Fate Core (and other recent Fate games such as Dresden) actually do a pretty good job of providing a suggested "menu" of powers and stunts for players to take; show them to your players and let those inform other stunt ideas through play, rather than trying to get the characters completely nailed down in advance the way you would have to in D&D.
To address the specific problems you mentioned:
The "blind sniper" problem is that in Fate, it's very easy for a cooperative party to stack multiple temporary aspect bonuses on a situation, then tag them all for a single super-successful roll regardless of innate skill.
Consider a sniper with a base skill of +0 - "has no idea what he's doing". First he hides on high ground (creating aspect "On a Grassy Knoll"). One of the other players - a tech - has made, and gives him some "Precision Armour Piercing Ammo", with suitable aspect. Then another player jumps into the road to stall the target into "Standing Still for a minute", so the sniper can create advantage by taking an acting to put him "Centred In My Sights".
Then the sniper free-tags all of those, for a base roll of +8 and a near-certain hit - with fate points to spare if he needs them.
Stacking enough of this sort of this can greatly reduce plausibility, but it's obvious behaviour for a group of D&D players where it's an expected part of the system to need to stack every combat advantage you can generate. I don't, however, feel it's as much of a problem as some Fate players do - this is behaves-as-designed. Fate characters are supposed to be able to beat pretty much any single obstacle if they can generate a convincing narrative. "He's not that good a shot, but it worked because an entire team was helping him take it" is a pretty good narrative to me, and gives the plot and characters room to develop further.
The "constant compels" problem arises when GM and players get caught in a Fate-point loop of constantly compelling aspects to force behaviour or non-actions from each other.
Remind your players, and remember as GM, that the point of compels is to implement narrative development. The Fate point economy will naturally limit how far players can push this, so it's not a problem unless the GM gets sucked into constantly offering Fate for compels. So don't. Compel when it's a narrative or character development to do so, not just because it's possible.
Long combats occur because groups have trouble actually taking out opponents of similar skill levels. This is particularly likely if your group is used to D&D, and trying to "wear them out" by all attacking individually. In Fate it's much more effective to set up a narrative about a couple of major attacks, taking multiple actions to support each other and make those go off well.
The best counter to most of these issues is threefold:
Let them have their moment. Players being able to tear through obstacles like this is not a problem in Fate the way it would be in D&D; it's expected behaviour of the system. Don't make it impossible, but ask what happens next and generate new obstacles. The problem can come more from a D&D-oriented GM feeling he has to make tasks "possible to fail" than from actual issues.
Your enemies are not idle. Don't GM in direct opposition to the player's actions, but instead change the game. Fate uses the same mechanics for social and combat conflicts, and it does so for good reason, expecting them to intermingle. Use it. Enemies will attack on social fronts, run away if they're in trouble, consult allies, adjust schemes, and all the other narrative options that are not often available to an orc in a dungeon.
Remember that in Fate, the player is always an informed participant, even when the character isn't. The gap between player and character knowledge is much bigger in Fate.
Situation: A hostage is tied to a chair. There is a trap which incinerates the room if anyone touches the hostage.
D&D answer: Tell the players nothing. Make hidden Perception checks. If anyone explicitly searches, make Search checks. Lie unless the players succeed at these checks.
Fate answer: Tell the players immediately that there's a trap. Let them make rolls and use aspects freely to see if the characters spot it. If one of the players has a "reckless" or "rescuer" aspect, compel it - offering a fate point if their character runs in carelessly and sets the trap off. Make the players partners in decisions that hurt their characters.