After delivering the Dragonstone to Farengar and completing the Bleak Falls Barrow quest, you unlock the next main quest of Act I: Dragon Rising.

If you ask Farengar about your reward, he will refer you to Jarl.

Open your quest journal.

Dragon Rising
I recovered the dragonstone from Bleak Falls Barrow, as the Jarl of Whiterun Requested. I should speak to him about my reward.

Talk to Jarl Balgruuf

As you are about to go speak to Jarl, Irileth barges into the room. She tells Farengar he needs to come with her at once. A Dragon has been sighted nearby. She tells you to come as well. Farengar is excited about the dragon and has a lot of questions. Where was it seen? What was it doing? But Irileth tells him to take it more seriously. If a Dragon decided to attack Whiterun, she doesn't know how they could stop it.

Follow Irileth and Farengar.out of the room and up the stairs. Jarl is interviewing a guard who witnessed the dragon from the western watchtower. He says he saw the dragon coming from the south, and that it was fast, faster then anything else he has seen before. Jarl asks the guard what the dragon is doing, whether it was attacking the watchtower. The guard replies that the dragon is not on the attack, it is just circling the tower overhead. Jarl tells him that he did good work and that we will take it from here, and sends the guard down to the barracks for some food and rest.

Jarl turns to Irileth and tells her to gather some guardsmen and get down there. She has already ordered her men to muster near the main gate. Jarl now turns to you. He needs your help again. He wants you to go with Irileth and help her fight the dragon. Because you survived Helgen, you have more experience with dragons then anyone else here.

He hasn't forgotten the service you did for him in retrieving the dragonstone for Farengar. He has instructed Avenicci that you are now permitted to purchase property in the city, as a token of his esteem. He also gives you a gift from his personal armory.

Farengar wants to join you and see the dragon as well, but Jarl refuses. He cannot risk both of you. He wants Farengar to stay in the keep and find a way to defend against the dragon.

If you open the quest journal,  you will see the quest has been updated:
A dragon has been sighted at a watchtower near Whiterun. Jarl Balgruuf has sent me along with his housecarl Irileth with reinforcement.
Your new objective is to meet Irileth near the western watchtower.

Follow her down the stairs, and onto the door. Exit to Whiterun. Keep following Irileth to the Western Watch Tower.

When you reach the main gate, Irileth will be briefing the guards. A dragon has been attacking the west watchtower. The guards are hesitant since they don't know how they can fight a dragon. But Irileth instills courage in them, telling them they are bound by honor to fight it, since it is threatening their homes, their families. Can they call themselves Nords if they flee from this monster? Are they going to let Irileth face it alone? But there is more then the honor at stake here. This is the first dragon seen in Skyrim since the last age. The glory of killing it is ours. The guards are now motivated and you all move out.

Follow Irileth to the gate and exit to Skyrim. Keep following Irileth and the guards. After a long run, you reach the western watchtower, which is running aflame. There are no signs of any dragon right now, but it sure looks like it has been here. You have to figure out what happened, and if that dragon is still skulking around somewhere. Spread out and look for survivors. You need to know what you are dealing with.

Approach the tower and look around. Suddenly the dragon will be spotted coming your way again! Irileth will instruct the guards to take cover and make every arrow count. Your objective now is to kill the dragon!

The dragon, Mirmulnir, will now strike. Attack him with all you've got! At some point the dragon is at half health, he will fly away but them come back and strike again, this time with a lot of firebreathing, so watch out. Eventually, you and the guards manage to kill the Dragon.

In your quest journal, it will show: The dragon attacked us and we killed it. We should investigate its corpse to see if we can learn anything about where it came from.

Your new objective is to investigate the dragon.

As you approach the dragon's corpse, it starts glowing as if it is about to erupt. Irileth will instruct everyone to step back. The dragon will soon expel his body and soul leaving only bones. You will absorb the dragon soul. Now you can use the Shouts section of the Magic Menu to equip your Unrelenting Force Shout. Using Unrelenting force, your voice is raw power, pushing aside anything - or anyone - who stands in your path.

A whiterun guard will approach you telling you that you are Dragonborn. Ask him what he means. He will tell you that in the old days of Skyrim, back when there were still dragons, the dragonborn would slay the dragons and steal their power. That is what you did, absorbed the dragon power.

You can tell him either that you don't know what happened to you or that he may be right. He will tell you there is only one way to find out. Try to shout, that will prove it. According to the legends, only the dragonborn can shout without training, the way the dragons do. Equip the shouting power from the magic menu. Follow the onscreen prompt to release a shout ("Z" on the PC). This will complete the optional objective of using your new shouting power.

In the quest log, it now says: When I approached the dragon's corpse, I absorbed some kind of power from it. I should return to Whiterun and tell the Jarl what happened. Your objective now is to report back to Jarl Balgruuf.

Before heading back to Whiterun, make sure to search the dragon's corpse as it will have a lot of loot on it. Then you can bring up your map and fast travel to whiterun. Near the main gate you might see a group of Alik'r arguing with the guards, which lets you pick up a new sidequest, "In My Time of Need".

Open the map again and fast travel to Dragonsreach. Enter the Jarl's court.

Approach Jarl's throne. He will ask you what happened in the Watchtower, if the dragon was there. Tell him that the watchtower was destoyed, but that you killed the dragon. He will say he knew he could count on you and Irileth, but there must be more to it then that. Tell him that it turns out you may be something called "Dragonborn". He will ask you what you knows about the Dragonborn. Tell him that when the dragon died, you absorbed some kind of power from it. He will say then that it is true, the greybeards were really summoning you, Ask him about the Graybeards. He will say they are masters of the way of the voice, living in seclusion high on the throat of the world. Ask him what those Greybeards want from you. Jarl will tell you that if you really are Dragonborn, they could teach you how to use your gift. When you killed that dragon, it revealed something in you and the Greybeards heard that. You have been summoned to the Greybeards at High Hrothgar, and you cannot refuse a Greybeard's summon.

Jarl thanks you for the service you have done to your city and names you Thane of Whiterun, the highest rank thats within his power to grant you. He assigns you Lydia as your personal housecarl, and gives you a weapon from his personal armory. He will notify the guards of your new title.

This completes the quest "Dragon Rising" and unlocks the next quest in the main story, "The Way of the Voice"