Why precession is the least intuitive thing in basic physics

Why a pushed gyroscope moves ninety degrees away from the push.

A pushed gyroscope moves not in the direction it was pushed but at right angles to it, because the push adds new angular momentum in the direction it was applied, and that new angular momentum has to combine with the huge existing angular momentum already pointing along the spin axis, with the combined direction ending up rotated ninety degrees from where the push alone would have sent it.

What is really going on

A spinning object already has a large angular momentum pointing straight along its own spin axis, simply because it is spinning, and pushing on that object to tip its axis over is really an attempt to add a new, smaller amount of angular momentum pointing in the direction of the push. Angular momentum, once added, does not replace what was already there, it combines with it, and the direction of the combined total is not the direction of the small push alone, it sits somewhere between the original huge amount and the small addition, tilted only very slightly from where it started, but tilted at right angles to the push rather than in line with it. This is the part that resists intuition, since a normal object simply moves in the direction it is pushed, but a spinning object's axis moves sideways to the push instead, because what is actually being redirected is not the object itself but the existing angular momentum it already carried, and redirecting an existing quantity by adding a new one at right angles to it moves the total at right angles to the addition, not along it. Precession is the name for exactly this sideways drift, and it appears in anything spinning fast enough for its own angular momentum to dominate whatever small push is being applied to it. Nothing about the underlying rule is exotic, it is the same combining of angular momentum in different directions that explains a spinning top's steadiness, only now applied to a push instead of gravity's steady pull, which is why the two phenomena, a top standing upright and a gyroscope swerving sideways under a push, are really the same mechanism seen from two different angles.

The bicycle-wheel comparison

Holding a bicycle wheel by its axle with both hands, spinning fast, and trying to tilt it downward at one side produces one of the most reliable surprises in basic physics, because instead of tipping down the way a push would tip anything else, the wheel swings sideways instead, turning in a direction nobody pushed it in at all. The hands feel a genuine resistance to the tilt, and the wheel's actual response, swinging around rather than tipping over, is the same right-angle redirection every spinning object undergoes when something tries to tip its axis, made obvious here because the wheel is large, heavy, and held directly in the hands rather than glimpsed for a moment on a spinning top across a room. Trying to fight the wheel, forcing it to tip in the direction originally pushed rather than letting it swing, takes noticeably more effort than the small original push would have suggested, because the hands are no longer just tipping an axis, they are actively overriding the wheel's own resistance to being redirected.

One figure worth keeping in mind

The faster the wheel spins, the smaller that sideways swing becomes for the same push, since a larger existing angular momentum needs a proportionally smaller redirection to absorb the same added push, which is the same relationship that makes a fast-spinning top wobble only slightly under a nudge that would send a slow one swinging wildly off to the side rather than merely leaning.

Where this stops being true

Stop the wheel spinning and the whole effect vanishes immediately, since precession only exists because there is a large existing angular momentum for a push to redirect, and a stationary wheel pushed the same way simply tips over in the direction it was pushed, the ordinary way any object responds to a push, with nothing sideways about it at all. A wheel spinning very slowly sits somewhere between the two extremes, tipping partly in the direction pushed and swinging partly sideways, which is often the clearest way to actually see precession happen, slowly enough to watch rather than fast enough to only feel as a strange resistance in the hands. Somewhere between those two speeds sits the point where an ordinary push stops behaving like a push at all and starts behaving like the strange sideways nudge described here, and finding that point by hand, slowing a spinning wheel gradually and feeling the response change, is one of the more memorable ways to meet the idea for the first time.