Serenity Rules Primer
Basic Mechanic: You succeed at something if [Ability die]+[Skill Die] >= [DC].
At character generation, you use a point-buy system to purchase the level of die you use for a particular attribute or skill.
You start with 48 Attribute points and then buy Assets (Things like Allure, Born Behind the Wheel, leadership, Linguist, etc) and Complications (Things don't go smooth, paralyzed, superstitious, deadly enemy, etc.). Assets cost 2 or 4 points, depending on the level of benefit, and taking a Complication will get you 2 or 4 points depending on their severity. You must purchase at least one Asset and one Complication, but may have up to 5 of each.
Then you purchase an ability die for each of six abilities: Agility, Strength, Vitality, Alertness, Intelligence, and Willpower. It costs 2 attribute points to buy a d2, 4 points for a d4, etc., up to 14 points for a d12+d2.
Initiative is derived from Agility + Alertness. Life Points come from Vitality + Willpower.
Then purchase skills. You get 68 skill points. You can buy general skills up to the d6 level only: 2 points for d2, 4 for d4, and 6 for d6. Then you can specialize in a skill, buying a d8 in the specialty for 2 points, a d10 for 4, d12 for 6, or d12+d2 for 8. [Thus, you could buy Guns d6, Pistols d8, and Rifles d8 for a total of 10 points.]
You get 1500 credits to buy gear (unless you take an asset or complication that changes that total)
The games uses plot points as rewards for doing cool stuff. You can spend them either prospectively on a roll to purchase extra dice (1 pt for a d2, 2 for a d4, etc), or to change the result of a roll retrospectively (1 point per point of change). You start with six and can keep up to 12 on hand.
Combat summary from FireflyRPG.com.
Character Sheet also from FireflyRPG.
A fillable version of the same character sheet (with menus containing the official Assets and Complications, for example) is attached.
[More to come]