New main objective: Locate the dragon burial site
If you want to head to Kynesgrove on your own and not accompanied by Delphine, you will have to dedicate some time to plan your journey. Open the world map and you should note that your destination is in the eastern part of Skyrim. The nearest big town is Windhelm (screen above) and if you have already visited it before (or Kynesgrove itself), you should of course consider using fast travel. Otherwise, you will have to decide whether you want to go there on your own and use the road signs, or use a carriage (e.g. the one in Whiterun Stable) for a small charge. After getting to Windhelm, you just need to head south and you will reach your destination.
If you want to head together with Delphine, you need to exit the Sleeping Giant Inn and follow her. Try not to get too far from her, though she should stop if you get too far way. Of course you will probably come across some Bandits and wild animals on your way (screen above), though eliminating them shouldn't cause you any trouble. The only exception is the Bandit by the Valtheim Towers, as you will be able to choose between paying and using Speech to try to persuade or intimidate him. Delphine will be heading towards Windhelm and turn south after reaching its borders, towards Kynesgrove.
Regardless of whether you reached Kynesgrove with Delphine or met her there, you will have to talk with Iddra (screen above), who will tell you that the Dragon has appeared beside the nearby burial site. Run across Kynesgrove, heading towards the new destination east of the village. After getting there, stand beside Delphine and listen to a "conversation" between two dragons, Alduin (the one who attacked Helgen in the beginning of the game) and Sahloknir. The latter one is the more important for now, as you will have to fight him.