On this page of the guide for TES V: Skyrim we have presented a walkthrough of the initial part of the main quest A Blade in the Dark. The introduction for this task is simple, just meet with Dolphine in Sleeping Giant Inn.
New main objective: Talk to Delphine
Note! Id' recommend beginning this quest only after completing Horn of Jurgen Windcaller, that is after heading with the Horn gained from Delphine to the Greybeards.
Head to the eastern room of the Sleeping Giant Inn (screen above) and lock the door behind you, as Delphine wishes. Wait for the woman to use the secret lever, unlocking a passage to the basement. Go down the stairs.
Approach the table beside which Delphine is sitting and initiate a conversation with her (screen above). The woman will inform you that she has been searching for the Dragonborn - who, according to her, lives to kill Dragons by absorbing their souls - for a long time. The woman will also tell you that Dragons keep returning to live throughout whole of Skyrim and the next "resurrection" is believed to take place beside an ancient burial site around Kynesgrove. Delphine will want to head there to slay the newly born Dragon. In the end you will have to decide if you want to head on your own and meet Delphine there, or go together. I'd recommend the latter option, unless you have already discovered Kynesgrove (or the nearby Windhelm) are able to use fast travel.