Generating electricity: Does wire thickness for magenetic induction matter?
The thickness of a wire directly impacts the resistance per unit length. Resistance (when current flows through it) causes voltage drop. Other than that, the thickness of a wire has no noticeable effect upon the voltage induced in it due to a changing magnetic field.

if I want to be sure I have a greater voltage output or whatever (with less resistance or impedence), do I increase the thickness of the wire or decrease the thickness of the wire?

Current flowing through a resistance causes a) a voltage drop, and b) heat. Neither of these things are usually wanted in a device such as a generator. Therefore your wire should be thick enough to handle the expected current with minimal, or at least acceptable, resistance. A thicker wire will have lower resistance.
It should be noted that, as is often the case, there are diminishing returns from ever increasing the thickness of the wire. If the efficiency of your device is already, say 95%, it may or may not be worth it to increase the thickness of your wire to increase your efficiency to 96%. Besides the fact copper costs money, thicker wire also adds bulk. When wires get too bulky to fit in the space that is allotted to them, one may need to use a larger structure to hold those wires. In transformers, for example, once one gets to the point where the number of turns of wire can no longer fit in the &quot;window&quot; within a core, one has to shift to a larger core. So, use wire that is thick enough to handle your current with acceptable losses, but don't go overboard, because the payoff may be less than the cost.
On a side note, you wrote:

I also know that when conducting electricity from a source to and end use that the longer the distance the more thicker the wire needs to be or else eddie currents and resistance will cause the wire to overheat (and thus either melt or cause a fire hazard).

Wire size has nothing to do with eddy currents, and you may want to use wire that is larger than the minimum needed to prevent fire. Or not. It depends upon the losses in your wire, and whether they are acceptable.