There are indeed many awesome hobbies listed so far by responders.
For my own:
I have recently taken up running. Which I find very enjoyable. My favourite is getting out at just before dawn to a trail that is near my house. Many times I can run and not see another person till i leave the park and make my way along the street back to my house.
I have no real recommendation for sports, as it would be very individual depending on the gifts or restraints on the aspie involved.
As for hobbies, again a very personal one. Just something that you enjoy that doesn't evertax your insequrities (if you have any.) Of course, this could be stretched to say if you have a hobby that is a slight stretch it give you a permanent (and at least slightly enjoyable) reason to confront any areas of your life that you may find difficult.
As an example, I REALLY enjoyed jazz and swing music. So I took up swing dancing to get used to human contact. It is something I still dislike, but it has conditioned me to it. There are many others of course.
A person with poor interpersonal skills could take up drama, someone with poor hand eye coordination could take up tennis. Some stepping stones in between now and then may be necessary of course, but food for thought.