Before the Storm must be completed in order to unlock the third main-quest (Bleak Falls Barrow) as well as a side-quest named The Golden Claw.
Some locations you have to visit during this quest are Riverwood, Alvor and Sigrids House, Hod and Gerdurs House, Whiterun, Dragonsreach.
You will meet Jarl Balgruuf the Greater of Whiterun, Proventus Avenicci, Ralof, Sigrid, Sven, Alvor, Dorthe, Frodnar, Gerdur, Hadvar, Hilde, Hod and Irileth.
You will start this quest when you exit the cave from the previous quest.
Watch the cutscene showing the dragon that saved you from the execution, and then talk to the companion that helped you escape (Hadvar or Ralof).
When the dialogue ends you will receive the first objective of this quest:
Objective 1: Talk to Alvor/Gerdur in Riverwood
NOTE: The name of the person you have to meet in Riverwood depends on the companion you picked during the first main-quest.
If you wish, you can follow your companion to Riverwood, or you can get there alone. Additionally, you can travel to Whiterun and talk to Jarl Balgruuf; but to complete this main-quest it is a good idea to follow your companion.
NOTE: It is important to know that in Skyrim you will witness a Civil War between the Imperials and the Stormcloaks. Since Hadvar is an Imperial and Ralof a Stormcloak, the quests you have to complete for them (or any other characters) may affect the main-quests.
Start exploring the region and follow your companion down the hill. He offers useful information regarding the faction you joined and he will guide you all the way. Eventually you will be attacked by several wolves.
Take them out and loot their corpses if you wish.
NOTE: In this walkthrough, we will not insist on how to explore the world of Skyrim; however, we will point out every area of interest that must be visited to complete all quests. Since Skyrim is an open-world videogame, it depends on your play style and on how much time you have at your disposal, to play a video game. Make sure you keep your eyes on your compass and always investigate all primary locations. Try to pick up all ingredients and flowers that grow nearby you and perform as many actions as you can in order to increase your characters skills.
On your way down the hill, make sure you activate the Guardian Stones near the main road.
Another primary location that you should visit is the Embershard Mine, but be careful because there are several thieves inside it.
When you reach Riverwood, you will participate in a series of dialogues between your companion and his relatives.
If Hadvar is your ally, you will meet his uncle (Alvor), Sigrid (Alvors wife) and Dorthe (Alvors daughter). If Ralof was your companion during this quest, you will meet Gerdur (Ralofs sister), Hod (Gerdur husband) and Frodnar (Gerdurs son).
Both dialogues lead you to your next objective and you will receive a gift. If you traveled side-by-side with Ralof, you will also receive a key to Gerdurs house, where you can rest.
NOTE: Make sure that you take all gifts, and sell what you do not need. Before leaving Riverwood, you should talk to all characters you meet. Sleep if you need to recover and dont forget to visit Alvor because he is a blacksmith that has useful tools you can use. You should also visit the Alchemy Lab located inside the Sleeping Giant Inn. Another area of interest is the Riverwood Trader, where you can sell your goods and talk to Camilla Valerius or Lucan Valerius, who will give you a new side quest called The Golden Claw.
Now, you are ready to travel to the Fortress City of Whiterun, in order to complete your next objective.
Objective 2: Talk to the Jarl of Whiterun
When you exit Riverwood, cross the bridge and go up on the hill, following the road with the river to your right.
Eventually you will reach two bridges. Go left and cross the Honningbrew Meadery and then you will hear the sounds of a battle near Pelagia Farm.
If you wish to start the Companions Quests, help the soldiers fighting the giant.
When the fight is over, continue your journey up on the hill until you reach the gate of Whiterun.
A guard will stop you, and at this point, you can persuade, bribe or intimidate him. Starting a fight is not an option.
Enter the city and explore it. You have many things you can see inside Whiterun, and each character you talk to, will offer useful information regarding the world of Skyrim.
When you are ready to complete the main objective, go north and cross the Kynareth Temple. Follow the stone stairs that will take you to Dragonsreach and approach the fire you see in the center.
Talk to Irileth and tell her why you want to see the Jarl. When the dialogue is over, the Jarl will want to see you.
During the dialogue with the Jarl, make sure you mention Alvor or Gerdur.
Proventus Avenicci will join the discussion telling you that if the Jarl decides to help Riverwood, the Jarl of Falkreath may get angry.
Eventually, the Jarl Balgruuf will send the troops and you will complete the main objective and the second main-quest of this act. You will also receive a small gift that will increase your armor skill and you will unlock the third quest, called Bleak Falls Barrow.
NOTE: Even if you can attack Jarl Balgruuf and his court, starting a fight while visiting Dragonsreach, is not a good idea. The Jarl cant be killed and you will be defeated. If you attack the Jarl, or his court, you have to wait three days before you can revisit this location.