I have 2 comet goldfish.  One is 8 inches long, the other 6.  I already know that the tank we have for them is much too small, but due to space restrictions in my home, we can not get a larger one.

Recently the older (8 in) on has been acting strange.  He floats towards the top of the tank, and stays there for hours, motionless.  The younger one began to follow suit.

This morning when I fed them, they were both floating at the top, strangely. But when I fed them, they began to move.

A few hours later, I saw the older one floating on his side, in the corner, breathing heavily with swollen, red, gills.

I freaked out, and took him out and put him in fresh water.  His progress has been improving a bit.  He's swimming around for a few moments, then returning to floating, sideways on the surface, breathing heavily.  There was also a white residue on the surface of the new water after he had been in there for about 10 mins. So I took him out and put new water in.

Is there anything I can do to save my fishy? He's a part of our family and we would hate to lose him.  I plan on cleaning the tank and replacing the gravel and everything.  I removed the other fish from the tank too.

If you put your sick fish into freshwater, i dearly hope that you made sure it wasn't strait from the tap. Most of that water is treated with chlorine and fluoride, the former of which can kill fish (humans too, in large enough doses, but you wont suffer any health problems by drinking tap water) you can make tap water pure enough for fish by adding some water purifier-the one i use is Tetra.
as to your fish...
it looks like he has a severe bacterial disease. i cant pin a name on it because there are a few with the symptoms you mentioned (heavy gills, white residue on water) however, i would suggest feeding your fish very finely mashed green ball peas as those will clear out any nasty stuff blocking your fish's intestines (if there is any).
i disagree with Tiam- swim bladder problems are much more noticeable that your description because fish have either a swollen area between the start of the tail and middle of the dorsal fin or a collapsed area in the same place. (either an over expanded bladder or a collapsed one)
i do agree that it could be ammonia and nitrite levels being through the roof. goldfish are notoriously messy fish, and if you have two big ones in a small tank, it would not surprise me if your levels were too high. test pH, but i doubt that has anything to do with your problem. if pH is too high or too low, it will kill a fish in 20 minutes.

my plan of action- run to your nearest fish store and get something called "Melafix". its an all purpose antibacterial remedy for all kinds of fish, freshwater or salt and any fish place worth your time will have it. also, buy some "Stress Coat"-that will stabilize the water of your tank. keep your other fish far FAR away from your sick one in a tank big enough for it (should be at least 10 gallons in the constraint of space, by the sound of your other tank) also, Aquarium Salt is a well known remedy to improve gill function. all aquarium places have this.
I hope that your fish survives and that your other lives a long, happy life with it and that you get a big enough tank for them eventually. any other questions you have you can post in this forum- please tell us the outcome.