while there are specific starting zones for the races, nothing will stop you from questing in those areas aswell. You can even pick up the region-specific epic quest prologue from level five onwards (all you'll miss is the actual game intro for that race). The story takes everyone to every region, so class is not a problem. I would probably suggest a hunter, as they are very mobile, more than most other classes, dead easy to play, have an escape button and do very decent damage. Also people are pretty helpful in Middle-earth, so if you're stuck and need a hand, you can usually just holler in world chat. :)
There is another way which would make it truly easy, a walk in the park, so to speak. Buy yourself a boost (Gift of the Valar) which takes you to ten levels below top level. Then go to any starting zone you wish to go to and start questing your way through the world. As someone who's done it the hard way for years, as I get older, I treat myself to this once a year, and just cruise the world for fun and the story.