Battery to AMP should be a minimum of 8guage. If you're running more than one amp, then 4 guage with a power distribution block using 8 guage to the amps is good. Amp to Sub 12 guage. Remote to amp 14 or 16 guage is fine. It just turns it on, doesnt' actually power the amp. None of those would cause an AMP to catch fire or overheat.

What would cause it is overheating caused by clipping. Someone else here mentioned it. You are getting clipping based on one of two things. The OHMS the amp is seeing is too low to remain stable at high volume levels, or the input gain on the amp is set too high which is causing the amp to be pushed beyond its limit. After you figure out what your ohm load is you need to adjust the gain on your amp to right level. If you don't have a volt meter handy, do the following.

Turn your system on. Turn the volume on the head unit to 3/4 maximum volume. Adjust the gain on the amp to maximum (it will sound like crap) then start to turn it down until you hear no distortion. Overheating of an AMP is not caused by small guage wires (unless it's the wire that's toasty).

 

I'm not an audio specialist, call crutchfield (www.crutchfield.com) and ask them what wiring package you need for your amp. Then if you wanna be cheap just look up the wiring package and see what size the wires are and then go buy the wires from somewhere else. You need at least 4 or 8, probably 4, guage wire from the battery to your amp. From the amp to the subs you might be able to get away with 16 guage. What are the ohms on your subs? Are they SVC or DVC?