Obviously you have no choice in escaping Helgen but what do you do for the next few days before you kill your first dragon?
I used to go straight to Riverwood get the "golden claw" mission, help out Faendel with his love life, go to Embershard Mine, then head up to Bleak Falls Barrow. Once that was completed I would clear out another unmarked bandit camp above Riverwood and then head off to Whiterun in hope of getting there in time to help kill a giant. In whiterun I would join the Companions, see the Jarl, speak to a Redguard about his sword and get a couple of jobs to kill some bandit leaders before speaking to the Whiterun court mage and starting the whole killing a dragon section of the game.
Now of course I do things a little different. I still go to Riverwood and enlist the help of Faendel, go to Embershard and clear the unmarked bandit camp, but I also clear out the other unmarked camp as you come out of Helgen and walk a bit around the shores of Ilinalta Lake, before heading off to Whiterun. I leave the Companions for a bit later now, although I've just started a new play through and haven't left Riverwood yet so I may try and complete the companions before going up to Bleak Falls Barrow.