The full rules are here, but here's the short version.
The Basics: The normal thing you'll be doing is taking your skill and adding the result of 4dF (a dF is a die that has sides -1, 0, and +1, so this will average to 0). So if you have a Great Athletics and get a total of +2, your result is Fantastic. Like you might expect there may be other bonuses or penalties to the roll.
Aspects and Fate Points: The main way to improve your die rolls is with the use of aspects. Every character has ten aspects listed on their sheet, but everything of significance in the world has aspects associated with it — NPCs (thugs could have Greedy), items (a sword could have Sharp), the scene (outdoors could have Strong Breeze), etc. If you want to improve a roll, you spend a fate point (of which you start out with five, and get more at the GM's whim or when your aspects are compelled (see below)) and invoke a relevant aspect ("I'm trying to break open this door, so I'm invoking my Human Battering Ram aspect!"). At this point you can choose to either reroll the dice or get a flat +2 bonus — you'll usually want the +2 unless you rolled a -2 or worse. If this still isn't good enough, you can keep spending fate points and invoking aspects. Note that you can invoke any relevant aspects, no matter who they belong to ("I invoke the thugs' Greedy to aid in luring them over to this side of the room").
Your aspects can also be used to earn you fate points. If the GM (or you yourself) feels like an aspect would penalize you in a situation ("You try to stop on the edge of the cliff, but your battering-ram-like momentum carries you over!") you get a fate point. You can even force this kind of thing onto other people by spending a fate point, though they still collect a fate point for being penalized.
In addition to this, you can use aspects to state facts about the world: pay a fate point and declare that since you're Rich, of course the theater manager will get you the best seats in the house.
Things to do with Skills: Skills work in broad terms exactly like you'd expect: you might roll against a fixed difficulty, or go head-to-head with someone and the higher roll wins. In the latter case, the contest may be simple (single roll) or extended (multiple rolls). When competing with someone, the margin of victory is generally significant: it might determine how much damage you do in a fight or how completely you've wooed the guy's pet tiger away. Having established that, here are some more specific rules:
Stunts: Stunts are additional snazzy things you can do. Some stunts cost fate points to use, while others are free (and in fact many are always-on).
Stress and Consequences: Characters have a Health track and a Composure track, with a number of boxes depending on their Endurance and Resolve, respectively. When a character loses a physical/mental contest, they fill in the appropriate number of boxes. When all the boxes are filled in and the character needs to fill one more in, they're Taken Out: their opponent can narrate what happens to them. To avoid filling in boxes, you can instead take a consequence. Characters can generally have up to three consequences (per track): a mild, a moderate, and a severe. Taking a mild consequence reduces the damage from an attack by 2, and the consequence lasts a scene or two. Taking a moderate consequence reduces the damage from an attack by 4, and the consequence lasts a day or two. Taking a serious consequence reduces the damage from an attack by 6, and the consequence lasts until there's some serious downtime, probably until the end of the session. Note that consequences are aspects, so they can be compelled or invoked just like other aspects.
The Skill List: