You can go off on your own right away if you want, or you can follow the guy as far as Riverwood (getting a brief tutorial on The Standing Stones along the way) and meet his family in Riverwood who can give you supplies and direct you toward Whiterun for the next quest on the main story. Which family you meet of course depends on whether you follow Hadvar or Ralof into the keep during the Helgen attack. Each family will talk to you and give you equally interesting, but opposing views on the political situation happing in Skyrim. Ralof and his family support the Stormcloaks, whereas Hadvar and his family support the Empire. This doesn't impact on the rest on the game, It's just an interesting insight into the views of these people with regard to the war.