(15 minutes or less)
Screw the handles onto each side of the wheel's axle. You may have to remove the outer nuts to clear enough axle for the handles. You may want to put plastic spoke guards on the hubs first to protect your fingers from the spinning wheel.
If you have the eyebolt, drill a hole in the end of one handle for it. Mount the screw eye in the hole.
Hold the wheel by the handles while another person gets it spinning as fast as possible. Sit on the stool with your feet off the floor, and tilt the wheel. If the stool has sufficiently low friction, the stool should start to turn. Tilt the wheel in the other direction.
Get the wheel spinning, and then use the eyebolt in the end of the handle to hang the wheel from a hook mounted to the free end of a chain or rope. Hold the wheel so that the axle is horizontal, then release it. The axle will remain more or less horizontal while it moves slowly in a circle.
If you don't have a chain or rope, rest the eyebolt on your fingertips. Be sure to practice this before you try a demonstration. You will have to move with the wheel as it slowly turns in a circle.
A rotating bicycle wheel has angular momentum, which is a property involving the speed of rotation, the mass of the wheel, and how the mass is distributed. For example, most of a bicycle wheel's mass is concentrated along the wheel's rim, rather than at the center, and this causes a larger angular momentum at a given speed. Angular momentum is characterized by both size and direction.
The bicycle wheel, you, and the chair comprise a system that obeys the principle of conservation of angular momentum. This means that any change in angular momentum within the system must be accompanied by an equal and opposite change, so the net effect is zero.
Suppose you are now sitting on the stool with the bicycle wheel spinning. One way to change the angular momentum of the bicycle wheel is to change its direction. To do this, you must exert a twisting force, called a torque, on the wheel. The bicycle wheel will then exert an equal and opposite torque on you. (That's because for every action there is an equal and opposite reaction.) Thus, when you twist the bicycle wheel in space, the bicycle wheel will twist you the opposite way. If you are sitting on a low friction pivot, the twisting force of the bicycle wheel will cause you to turn. The change in angular momentum of the wheel is compensated for by your own change in angular momentum. The system as a whole ends up obeying the principle of conservation of angular momentum.
Unfortunately, the gyroscopic precession of the wheel hanging from the rope is not explainable in as straightforward a manner as the rotating stool effect. However, the effect itself is well worth experiencing, even though its explanation is too difficult to undertake here. For more information, consult any college physics text under precession.