Is it possible to journey the world without the dragons hunting you? If I don't do any of the main quest after escaping Helgen can I avoid that?

I think they start once you turn in that stone you get from bleak falls barrow. As long as you don't do that you'll be fine. =)

The first dragon spawns during the main quest, the random spawns start after you kill that one. So yes you could go through most of the game without dragons, but you would also go through most of the game without shouts and several dungeons have secret areas and alcoves you need Whirlwind Spirit to get to (basically jump puzzles you have to notice as they are cleverly concealed within the dungeon terrain that allow you to get high level containers)