Here's a D20 adventure game

Basics and characters

Write down your character name and then a description. Make it concise or detailed. Include some skills or a calling/profession.
Roll 2d6 and add six. These are you Health Points, or HP. Write them in near the top. Hits wear them down. When you have no more HP, you cannot take a hit without serious consequences.

Rolls

Further down, you have three rolls: Fighting, Skill, and Magic. These have a base value of 6+.

For a random character, roll three d6 and assign each die to a roll.

For a bespoke character, distribute 11 points between the rolls, giving at least one point to each roll.

Now sort out what gear you have with your GM.

What would you use those rolls for?

Use that fighting roll in combat, to see if you hit something, or if you can avoid being hit. This also applies to physical challenges.

Use that skill roll to sneak, tamper, fiddle, search, and do anything associated with your craft.

Use that magic roll to cast magic, if that's in your power, resist magic if it comes to that, and otherwise recall fine points of lore.

Running adventures

In most cases, when characters are tested or challenged, choose an appropriate roll. If the character makes the roll, equalling or exceeding their number on a d20, they are successful.
Apply modifiers that fit the situation. Add points to the roll number to increase the difficulty, or deduct points to lower difficulty. A range from +5 to -5 should represent most conditions.

Experience

The system is intended for quick pick-up-and-play gaming, simply adapting any module or adventure or idea the GM has at hand. Hence, characters can begin with any adventure and level of expertise the GM requires.

However, if the game runs into campaign play, you can begin with an experience level of 0: adventurer. This can go up to 5: hero. The difference between a character's level and the challenge of a monster or task can be used as a modifier.

Running combat

Assume that a character's fighting roll represents the sum effect of their skill, weaponry, and armour matched against a middling foe.
Order of attacks and defence in each round of action depends on the tactical situation, so that's for the GM to decide, though generally the characters with better fighting rolls get to go first.
In each round, characters get a chance to attack and hit using their fighting roll.
The GM will decide if the foes they face have a "chance to hit" (a roll based on fighting) or simply hit automatically, as some powerful creatures will.
If a character is hit, then they can use their fighting roll as a save to try and block, parry, or dodge the impact, if their fighting style and the situation allows.

At the end of the round, deal damage to anyone who has been hit:

Light weapons: d4 (knives, staffs)
Skirmishing weapons: d6 (axes, daggers, darts)
War weapons: d8 (swords, arrows, spears)
Heavy weapons: d10 (great sword, pike)
Magical or terrible weapons: d12 (dragon claws, wraith-sword)

If a character has armour that is proof against a particular attack, then the GM can account for this by temporarily dropping the damage category (so a sword (d8) striking mail might do lighter (d6) damage).

Magic

Magic is tricky, so take your time over it. Magic effects each world differently. As a GM, you can't allow magic that would shortcut play, but magicians should be able to use their magic as effectively as other characters. 

Perhaps there's a price for gaining access to magic, such as a permanent reduction in HP or other rolls.

When you have worked out how magic works, assume that characters can effectively cast a handful of basic spells they know using a magic roll. Spells of greater power, or outside of the magician's ken, are progressively more difficult to cast.

Magic should be risky. If a roll fails disastrously, then inflict an interesting twist, side-effect, or even damage on the hapless caster.