Don't stay awake for 24 hours unless you're a) doing necessary school or Uni work, or b) with mates. Speaking from experience, doing it just for the hell of it is the most pointless thing you'll ever do.
But if you have to:
* don't watch TV. You'll fall asleep.
* don't lie down. You'll fall asleep.
* don't read a book. You'll fall asleep.
Music, if of suitable type and volume, can be helpful, as can computer games or anything you have to interact with. And keep the lights on.
24 hours is in fact not that difficult. Just have your normal day and don't go to bed. It's when it gets to 36 hours that it gets tricky.
Vaguely related - there was a guy a couple of years ago, who decided to live as if there were 6 days in the week and write a commentary on it on his blog. This meant that each 'day' had 28 hours, and he got up and went to bed at a different 'real' time each day. He was OK until he had to go to bed just as it got light - that finished him off.