You have a car battery sitting around somewhere that works?
If not you can connect it to the one under your hood. Take the yellow (constant power) and red (memory/acc) wires coming out of the back of the head unit, twist them together and touch those to the positive side of the battery. Then take the black wire and touch it to the negative side. That will allow the radio to come on if it is working.
If you need to test the speaker out puts, just get an old speaker (one out of the house would work in a pinch) and touch the green/green black, purple/purple black, grey/grey black, white/white black to the pos and neg of the speaker. Not all at once. The solid is positive and the black stripe is neg and each one goes to a different speaker (front right and left, rear right and left). I just don't remember off the top of my head which ones go where. I know the grey and white ones are front and the green and purple ones are rear but I can't remember which side.