Rules Introduction

Serenity is based on the Cortex system, has a point based classless skill system and uses dice (d2 through d12) to represent your stats and skills. You come up with a character concept and choose the skills that you think fit your character idea and background. There are no preset classes or archetypes. Stats are Agility, Strength, Vitality, Intelligence, Willpower and Alertness. D6 is average. With the dice system, your character stats might be Agility d10, Strength d8, Vitality d8, Intelligence d8, Willpower d10 and Alertness d8.
Skills start as a general skill which goes up to a d6, then splits into specialties at higher levels. So a Pilot like Wash would have Pilot d6, Shuttle d10, Astrogation d8, Transports d12. A gunman like Jayne would have Guns d6, Pistol d8, Shotgun d8, Assault Rifles d10.
There are also Assets (things you're good at) and Complications (things you're not). Wash has Born Behind the Wheel, which gives a bonus whenever he's flying. But he also has Lightweight, giving him a penalty to rolls against against getting drunk or resisting drugs and disease. Jayne has Fightin' Type, which gives him an extra non-attack action (such as moving or dodging) in combat, but he is also Crude, giving him a penalty in social situations or making Persuasion rolls. A lot of Complications are story hooks like Credo: Never leave a fellow soldier behind or Prejudice: Hates Alliance officials.
Skill checks are made by rolling your relevant stat die and skill die and adding them together. So if Jayne is shooting a pistol, he would roll Agility + Pistol Skill. If you don't have the appropriate specialty, you would use your general skill. So if Jayne fires a Submachinegun, he would roll Agility plus his Guns d6. The stat and skill are not locked pairs, the GM can have you make an alternate pairing based on the situation and the player can suggest an alternative if they have a justification to back it up. Gambling is normally a Willpower roll, but a player who has a highly intelligent character might say their gambling style is based on calculating odds and counting cards. Or they may be trying to cheat by dealing from the bottom of the deck and make a Dexterity roll. Shooting guns is normally Dexterity, but if your character picks up a machinegun and fires it from the hip, it could be Strength plus Gun skill to handle the recoil.
Skill rolls can be against a target number reflecting the difficulty of a task (diagnosing an engine problem, speeding down a busy street, climbing a wall) or opposed (shooting at a dodging enemy, following someone who is trying to lose you in a crowd). Bonuses are reflected in changing die sizes. A +1 step bonus raises your die size,changing your d8 to a d10. A -2 step penalty changes your d8 to a d4). Rolling a 1 on all your dice is a botch, so it's better to roll 2d6 than 1d12. Multiple actions are -1 step penalty for each additional action. Shooting three times is a -2 step penalty to each roll. There's no critical hit, but if you beat your target number by 7 or more, you get an exceptional success which will result in things like extra damage or story benefits.
Chips are awarded for good rolls and roleplaying and completing adventures (the system I use in D&D is derived from Cortex). Chips can be spent to add bonuses or cancel penalties to rolls, change lethal damage to stun damage or buy storyline events. In one of the first games I ran, the player characters wanted to knock on the bad guys' door and pretend to be drunks who couldn't find their hotel room, but they didn't have any booze. One of the players threw in some chips to say "I always keep a cooler of beer under the seat". If you try to buy a storyline event, the GM can decide you didn't spend enough and give you an issue. For example "I spend 1 chip, the bartender is an old friend of mine" And the GM says "One chip? Okay, he's an old friend, but you owe him money". Chips are meant to be spent frequently, but they are also your XP that you spend to raise your skills.
Damage (called life points) is tracked on a row of check boxes, lethal or wound damage is counted in from one end, stun damage from the other. When they meet in the middle, you have a chance to pass out. You can take all your damage in stun points and be knocked out (like in a fist fight) or all your points in wound damage and pass out from wounds. It's usually a combination of the two, so it's easy to be knocked out but not killed. If you take all your life points in wounds, then you have a chance to die. Weapon damage is a combination of damage from the weapon and the to hit roll. The amount you exceed the target number by is basic damage and is split between stun and wound (round up stun, round down wound), then you add the weapon damage. So if I need to roll a 7 and I get a 12, I beat the target number by 5 and do 3 stun and 2 wounds, then I roll a d8 for the weapon's wounds and get a 6, giving a total of 3 stun and 8 wounds. Called shots and exceptional successes can add extra damage; armor reduces it.