On this page of the guide for TES V: Skyrim you will find a walkthrough of the first part of the main quest Dragon Rising. This is not a very difficult part of the quest, as it involves a meeting with the Jarl of Whiterun - Balgruuf.
New main objective: Talk to Jarl Balgruuf
Follow Irileth, choosing the stairs leading to where Jarl Balgruuf is (screen above). During the conversation, you will be asked to join Irileth and the rest of the guards to head to where the Dragon was spotted and slay it. Balgruuf will also give you the prize for completing the previous quest, that is finding the Dragonstone in Bleak Falls Barrow, which will be an armor element (its type depends on your experience level and the chosen gameplay style; for example - a warrior using one-handed weapon should receive a shield). Jarl will also inform you that you can buy a house in the city and in case you would want to, you need to talk with Proventus Avenicci found in Dragonsreach and pay 5000 gold. Note - as you will be heading out for your first fight with a Dragon, you should consider visiting the traders in Whiterun to buy some better equipment (at best such which will increase your fire resistance), potions and other items.