As the title states, I am trying to save an image that I can see normally in the UV/Image Editor. I have done some compositing to it, and everything works inside Blender, but when I press F3 and save it to my desktop, the file that shows up there is blank, transparent. I have 3 render layers, all being affected by my composite nodes. In the layers tab at the bottom of the UV/Image Editor screen, I have Composite selected. It looks great inside of Blender, and I don't see why this isn't working. Any help would be much appreciated, thanks! If any extra info or screens are needed to try to help me solve the problem, please request!
The problem is that the image you are saving is entirely transparent except for the lights on the wings. You can't see this in blender because you have selected the RGB display mode, which disables the effect of the alpha channel. If you select the RGBA display mode in the header of the image editor, then the problem becomes clear:
As a quick fix, you could disable Use Alpha on the composite node and then save the image as RGB instead of RGBA.
However, in case you want an alpha channel, then the cause of this lies in your compositing setup (Be sure to select RGBA in the header here too, so you can see what's going on).
Mix/Add nodes don't handle alpha very smartly on their own, so you'll have to handle alpha for them, or use some alpha over nodes.
Also note that add nodes literally just add the values of all the pixels together. As a result your spaceships are coming out a few times brighter than they would be.
Here I've remade your compositing setup, working from the bottom (here the bottom is the background, which goes under all other elements) to the top (lens flairs/bloom effects generally go above all other elements):
Compositing things in this order makes a lot of this alpha handling unnecessary, and generally avoids extra headaches.