Retrieving the Dragonstone
Fast travel to Riverwood -- open your map, select it and then hit the fast travel button -- and if you need to go ahead and sell off the extra stuff you are carrying to the Trader here. Bear in mind that until you get enough money to buy a house -- and thus have someplace to store things -- picking up items you actually want to keep is not such a good idea...
Follow the pointer out of the town and up to the Black Falls Barrow which will get flagged on the map as a location you can fast travel to -- you will likely be attacked by bandits outside, so take them out and loot them, then head inside the first area and take out and loot the bandits there. You will need to leave and fast travel back to town to sell off the loot since you are close to being overburdened now.
Return and head inside, and make yoru way to the back, where you will battle another brigand and find a puzzle you need to solve -- three pillars that can be turned to show specific images and a level. If you look above the images for the first two are mounted on the wall - the middle one has fallen off but you can find it below near the lever, so you learn that the key is Snake - Snake - Fish. Put anything else in and pull the lever and you get a rain of poison arrows, so don't do that. Put the correct ones in and the door in the wall opens.
Inside there is a table you can grab some stuff from and read a book called Thief -- and to the left of it is a soul gem you should grab as it will come in handy later. There are also potions to be had on shelves as you progress through the area so be sure to grab those!
To your left is a corckscrew stairs that leads to the level below, and you should see three or so Skeevers (wild animals) come out of that and attack you -- so kill them! Now head down and you will find a table with a Fireball Scroll and a Weak Paralysis Potion -- grab those and you will hear someone yelling.
Obtaining the Dragonstone and Revealing a Secret
The Black Falls Burrow
Work your way to the blocked passage and smash it down to find a very large Frostbite Spider that you must kill -- be sure to grab the two bottles of venom from it, and then go have a look at what was going to be its dinner -- Arvel the Swift.
You ask him where the Golden Claw is -- remember the Golden Claw? -- he promises to tell you all about that, if you cut him down that is. Of course he runs and you have to chase him down and kill him as well as some undead. Loot them all then backtrack to loot the corpses in the spider room and grab a Common Soul Gem in the chamber he ran through. Do not forget to search any urns that are not marked as being empty -- you can get some good stuff from them!
As you continue along the path past where you dealt with the thief and the undead, mind the bloody spike trap yeah? You battle more undead and get past a swinging ax trap, then more undead and some lootables. When you get to the end chamber kill the tougher undead and then pull the chain to open the gate, and continue on. There is a chest you can loot in here just prior to dropping down into the next lower area.
-- The Sanctum --
Head to the left and you will find the entrance to the Sanctum -- go inside and work your way around, up, and over the bridge to reach another passage that will take you to a puzzle door. You will be glad that you got the claw from the thief earlier because it is the key to this door!
Check the golden claw to see the correct sequence -- they are engraved on it -- it is (top to bottom) bear, hummingbird, owl. Then use the claw to unlock and open the door!
Now head all the way to the top of the dias and the back to absorb the Power Word -- and learn that YOU are a Dragonborn! Remember that book in the first Keep I said you should grab? Now you know why!
After you absorb the Word of Power the Undead Boss climbs out of the crypt on the dias -- slay him and loot the tablet from his body! Now head up the stairs nearby and use the lever to leave -- fast travel to Riverwood to both to sell your loot and to complete the quest "The Golden Claw" for some XP and gold. Good on ya! Now sell the loot to him to lighten your load!
Be sure to level-up your skills if you have yet to do that -- you should be at Level 5 at this point, which is a good thing, as we needed to level to do the Sword Quest!