Gyroscopes-Specific Questions by an interested High Schooler

[Styker7]

Recently I've become fascinated with Gyroscopes and how they work. I am a high-schooler that only has a small physics background so any "dumb downed" answers would be appreciated.

First, I rigged up a bicycle wheel gyroscope (in which i hung a bicycle wheel from a rope and twisted the wheel to watch it turn in a circle about the rope). What I don't understand is how this example of a gyroscope pertains to a mechanical gyroscope. In the mechanical gyroscope the unmoving point is the center piece.
But what is the unmoving point in the bicycle wheel? (An analogy as an answer would be very helpful for me here. I.E. Bicycle Wheel=Center point; Rope=Gimbal)
How does a mechanical gyroscope show precession?
How is it that a mechanical gyroscope is amazing? I assumed that because it is not fixed to any of the gimbals it does not need to move in the direction that they are moving.
Also, pertaining to the bicycle wheel gyroscope; I understand that precession is caused by Newton's first law which, when implied, means that since the bicycle wheel can not progress in a straight forward motion it must proceed in a circular fashion. Is this correct?
How is it that precession cancels out the gravitational force?

 

[haruspex]

I'm not sure that I understand any of your questions exactly enough to answer them, but let me explain a common misunderstanding about gyroscopes.
Most people think that a gyroscope resists having its axis rotated. That's not quite correct. In reality, if you try to rotate the axis one way, the gyroscope will happily rotate its axis, but at right angles to both the axis and the applied torque. If holding it in your hand you naturally resist this tendency for it to move in a way other than you intended, and what you feel back is, in a sense, your own resistance.
However, mounted in gimbals you can observe the orthogonal compliance quite easily.

With your bicycle wheel, I'm unclear how you attached it to the rope. If you tied the rope to the spindle then the wheel would be free to turn without twisting the rope. In this arrangement, you could hold the wheel tilted at an angle to the horizontal and give it a good spin, about the spindle, directly. You should then see the wheel precess around the rope.
Gravity is exerting a torque at right angles to both the spindle and the rope. This results in a rotation at right angles to both this and the spindle, i.e. around the diameter of the wheel at greatest angle to the horizontal. But as soon as it starts to move that way, the torques change too, and the end result is precession around the vertical axis.