Before the Storm (Skyrim)

Background
"Hadvar suggested that I head to the nearby town of Riverwood. His uncle Alvor is the blacksmith there and should be able to help me".

"Ralof suggested that I head to the nearby town of Riverwood. His sister Gerdur is the mill owner there and should be able to help me".

Walkthough
You emerge into the bright light of Falkreath Hold, north of the still-smouldering ruin of Helgen. You and your companion watch as the dragon responsible for disrupting your execution flies away to the north. No matter whether you followed Hadvar or Ralof, talk to him and then make your way to Riverwood. You can follow your new friend down the hillside, but if you lose him simply meet up with Hadvar’s or Ralof’s contact in Riverwood.

On your way to Riverwood you will be attacked by some wolves, but that's nothing you cannot handle with the help of your companion. You will also find three Guardian Stones: The Warrior, The Mage, and The Thief. Pick one according to your playing style and continue following the path. Make sure to discover Embershard Mine. There be a single bandit standing guard who makes for an easy kill.

(OPTIONAL) Now, cross the river and you should notice a bit of a path, follow it to uphill to the north and off the west. At the end of the path, you will notice a set of stairs. These lead to an Iron Ore vein. If you continue straight where the path turns toward the stairs, you will notice a Treasure Hunter in a small clearing just shy of the snow (north of the waterfall closest to Riverwood). You can continue west to discover Anise's Cabin.

After your arrival in the village you will be led to a house depending on who you chose to follow in the opening quest. You will also be offered a gift. However, you can safely take all of the items offered without anyone complaining. Your Riverwood contact will also tell you to report to Jarl Balgruuf in Whiterun, which is your new objective.

Exit Riverwood to the north, follow the path with the waterfall rapids to your right (east), and begin a hill descent. Take the left (west) path at the two bridges, on the opposite side of the stream to Chillfurrow Farm, and pass the Honningbrew Meadery. Make your way past the Whiterun Stables and then up the hill, over the drawbridge, and to the main Whiterun gate. As you approach, a guard stops you; the city is closed with the news of the dragons spreading faster than you can travel. Simply tell him that Riverwood calls for the Jarl’s aid, the standard Persuade/Intimidate/Bribe options likely will not work here as your speech skill is probably too low. Once inside, head up north to the fortress towering the city, Dragonsreach, and enter through the main gate. The Jarl's Housecarl, Irileth, will ask you your business, but regardless of your answer you will end up talking to Jarl Balgruuf. You can lead the conversation with him in any way you wish, in the end the quest will be completed and you will be rewarded by the Jarl.

Rewards

 * Unenchanted Heavy or Light chest armor. The type depending on your level.