There's some weird physics going on in this. In "up the stairs" I was experimenting with some stuff... I put a yellow and a blue wheel on the ground, connected them at all four outer points (top to top, right to right, etc), put the red wheel behind, and connected that to the center of the yellow wheel.
After some time, the physics starts to go crazy, and the contraption somehow launches itself at high velocity into space!
Aha, here's the link:
http://FantasticContraption.com/?designId=676058
Some of the previous ones, I've pretty much just brute forced the wheels through, and dragged the red one along behind.

Though I admit I enjoyed making a catapult in that one level...
You could have added a wooded rod to connect the "axles", used two drive wheels, and had the red wheel ride on top:
http://FantasticContraption.com/?designId=679165
or two wheels work fine with just a little traction:
http://FantasticContraption.com/?designId=679786
Aha, finally past the stairs. My contraption just wouldn't make it all the way up... so I built another contraption to give it an extra push from behind.
http://FantasticContraption.com/?designId=676511