Angular momentum is a hard concept to teach in introductory physics courses. This is partly because it is a quantity known as a vector that has both a magnitude and a direction, and adding and subtracting and multiplying vectors is more complicated than with scalars, like mass. Other things like force and velocity are also vectors but in those cases the directionality at least is intuitively obvious. The direction of the force vector is in the direction along which the force acts and the direction of the velocity vector is the direction in which the object is moving.
But when it comes to the angular velocity of a rotating object, which is also a vector, the direction of the vector is given by something called the right-hand rule. To visualize this, curl the fingers of your right hand into a fist in the configuration that we call ‘thumbs up’. Then orient your hand so that the curling of the four fingers corresponds to the rotational direction of the object. Then the thumb will point in the direction of the angular velocity vector. So the hands of a clock, for example, rotate clockwise (duh!) but the angular velocity vector of the hands points directly into the plane of the face of the clock.
Why this weird system? It is partly a result of necessity and partly a result of convention. But it works extremely well.
Finding the angular momentum is a little more complicated. For an object that is symmetrical about the axis of rotation, which is what I will consider here, the angular momentum vector is parallel to the angular velocity vector. This is true in the case of a wheel rotating about the axle passing though its center.
Things get a little more complicated when we relate the change in angular momentum to the cause of that change, known as the torque. Take a look at this nice demonstration of the relationship between the direction of the torque and that of the change in angular momentum of a spinning wheel.
According to the laws of physics, the direction of the torque has to be in the same direction as the change in the angular momentum. Since a vector has both magnitude and direction, a vector can change just by changing direction without changing its magnitude, and that is what is happening here. The torque is causing the axis of the wheel to rotate. Since the direction of the angular momentum (and of the torque) is not intuitive, the direction of the change in it is even less so. It all works out nicely but it takes a while for students to really grasp it.
I have done a similar version of this demonstration using a spinning wheel and standing on a rotating turntable but I like this better for two reasons.
One is that he uses a motor to spin the wheel to a high rpm producing a much larger angular momentum. The other is that hanging the wheel at one end from the rope makes the direction of the torque easier to visualize and makes the effect more dramatic because it is visually arresting to see a heavy wheel suspended from a cord attached at only one end of the axle remain with its axis horizontal. This only happens if the wheel is spinning and it causes the axis of the wheel to rotate.