What is the purpose of the directions for vector quantities such as angular velocity and torque? Looking at just both angular velocity and torque, many other exhibit this same property. What is the purpose of and usefulness of having a direction that is in the Z-direction? (assuming that, in the case of torque, Force and moment length is x and y). I understand that since they are vector quantities they have to have both direction and magnitude, but I don't see the usefulness of the direction? I mean I can change the direction of torque depending if I choose counter-clockwise to be positive or negative? 
 A: "...I don't see the usefulness of the direction?"
A top spinning about the $\mathbf{\omega_1}$ direction just looks different than a top spinning about the $\mathbf{\omega_2}$ direction.  You need the direction  $\mathbf{\omega}$ is pointing to express that difference. 
"I mean I can change the direction of torque depending if I choose counter-clockwise to be positive or negative?"
Yes, if there were just one direction that torque could be about, you could express CW or CCW with just a plus or minus sign.  However, just like for $\mathbf{\omega}$, there are many directions that torque might be about, and you need the vector direction to tell them apart.
A: (Warning: non-formal explanation ahead.)
Any vector's magnitude and direction really serve two purposes: telling you how much "oomph" a quantity has (magnitude) and where the action is taking place (direction). Let's say you only have the torque vector. (Please reference this link).
Because torque is defined as the cross product of the radius vector and the force vector ($\tau = r \times F$), you know that the object is rotating on a plane perpendicular to this vector. 
If the vector is pointing upward, you know that the torque must be applied in a counterclockwise direction, and if the vector is pointing downward, you know that the torque is applied in a clockwise direction. This realization comes directly from the cross product definition of torque. 
You gain even more information from the length of the vector, as it tells you how much "oomph" the quantity has.
