If you take an aluminum wire and tap on one side, the other side will move. Is this a message that is transported faster than lightspeed?

No.  When you tap one side the other side does not move instantaneously. It appears to do so because our nervous systems are very, very bad measuring instruments. The disturbance is not carried at the speed of light. It is carried at the speed of sound in  Aluminium. Interestingly enough, the speed will depend on whether you hit the wire along its length or transverse to it length.  The speed of transmission of the disturbance will be equal to the speed of sound in aluminium if the disturbance is along the length. Otherwise it will be equal to the the speed of transverse wave in a aluminium wire and will thus depend on the thickness of the wire. EDIT: - I am not talking about sending sound through the wire. Every mechanical disturbance creates sound waves and all mechanical disturbances(if they are longitudinal ) will be propagated at the speed of sound in that medium(be it water, air or metal).

No, the information moving through the wire is moving at the speed of Sound, not Light. Aditya Nanda addressed all the issues I can think to mention, so you have pretty much all you need in that answer.

It will be transported at the speed of light: I'm going to assume that this is a perfect system, in that the wire in arranged such that all of the force you apply is transferred through the wire onto the other end, and that the wire is infinitely rigid, with no flexing (so the wire will have to be perfectly straight and rigid). Let's look at what is holding the atoms in the wire together, which are the electromagnetic bonds between the individual atoms of the aluminum wire. These electromagnetic forces are forces that operate through fields. Through the Maxwell equations (I'll save the details for you to explore on your own), it can be shown that electromagnetic fields and forces propagate at the speed of light. So in the situation you described, a person will push on one end of the infinitely rigid and perfectly straight wire, so that all of the force they apply acts on all of the atoms on one end of the wire, which through Newton's third law will apply a force to the next atoms, and so on. Since the forces between the individual atoms are electromagnetic forces, then the force from one end will propagate at the speed of light. But obviously there is no such thing as an infinitely rigid material, there will be some delay as the atoms in the wire move closer together (keep in mind the force will still react at the speed of light), which increases their potential energy to where they will want to move away from each other, therefore propagating the force from one end of the wire to the other. At least that is my logic with regards to your question, I may be wrong.