The Elder Scrolls V: Skyrim/Bleak Falls Barrow

Bleak Falls Barrow is the third quest in the main storyline. After you meet with the Jarl of Whiterun, you will be sent to speak with the court wizard, Farengar Secret-Fire. Although he is the official quest-giver, you are likely to complete this quest if you first encounter the sidequest of The Golden Claw, or if you decide to explore the barrow on your own initiative.

At low levels, you will likely encounter bandits and draugr near within the barrow. If you delay and instead arrive at a high level, you may also encounter a frost troll.

Approach
To reach Bleak Falls Barrow, you should first head to the tower that is to the east of the main barrow. Although you can approach it from a few directions, it may be easiest to approach it from the east by crossing the bridge over the river, and heading there on-foot. The tower itself is home to three bandits, who are easily defeated. Continue to the barrow, and you will encounter more Bandits on the outside.

Bleak Falls Barrow
When you enter, you will overhear two bandits talking about Avrel going on ahead. The bandits will attack when you approach, and they also guard a treasure chest which you can search.

Continue down the stairs. They will lead to the first puzzle room, where a bandit will most likely kill himself with the retaliatory trap. To open the door, you need to rotate the symbols on the left to match the symbols appearing above the door (although the middle display crashed to ground level). From left to right, the symbols are snake, snake, and whale. Flip the switch, and the door will open to the next room. Be sure to read the Thief skill book on the table before going down the stairs.

Arvel the Swift
You will find skeevers, and before turning the corner, you will hear Arvel calling for help. When you approach, the Wounded Frostbite Spider attacks. Kill it, and free Arvel by attacking the web.

Arvel will run away, and you will need to chase after him. He will head a few rooms ahead, right to a chamber where draugr rise and attack. If you didn't kill him before then, he will likely be defeated by draugur, or is guaranteed to be killed by a swinging wall trap. Collect the golden claw from the body.

The Golden Claw
Continue past the swinging wall trap, and the swinging blade trap. The next room had three draugr, and the destination leads upstairs. When you reach the room with the creek, head through the metal gate by pulling the chain to the right.

In the waterfall room, you can cross the bridge. If you want a treasure chest, you can head down to the bottom.

Continue to the golden claw door. The combination for the golden claw can be found in the inventory menu, and zooming into the object to get a better look. From top to bottom, the combination is a bear, moth and owl. Use the claw in the middle of the door to open it.

Word Wall
Once you're through the door, you enter the word wall room. You need to approach it, and you learn the first shout word, which you can use after completing the next quest.

Once you obtain the word, the last draugr will emerge (usually a Draugr overlord). Defeat it, and retrieve the dragonstone off its corpse. If it's overpowering, there is a ledge to the right of the entrance on the far side of the bridge, which you can use to snipe the opponent. After defeating it, you can retrieve its weapon, which has a frost enchantment. You can leave the barrow using the nearby exit.

Completion
After clearing the barrow, you can head to the Riverwood trader to return the golden claw for 500 gold.

Although returning the dragonstone is immediately harmless, the next main quest will start random dragon attacks. This can result in NPC questgivers from dying before they can give their mission, although "essential" NPCs will remain unharmed.

Head to Whiterun, and return to Farengar. You will see a short scene with Farengar in discussion with Delphine, and you can give the dragonstone when they're finished the discussion. Be sure to ask for the reward.

If you didn't report the dragon attack to the Jarl, you will have to speak with the Jarl first before you're sent to Farengar. The only overall difference is that you don't get to see Delphine discussing the dragonstone.