I've tried everything! I cannot get the tank to be pretty and crystal clear. 

I've changed the water weekly. Nearly 60 % over the past week. Changed the food. Cut back on feeding. There's nothing in the bottom except a cave, a coral-thingie, and 3 river rocks ( all purchased at the fish store).

There's no food lying around-it doesn't stand a chance with their appetites!
No poop lying about. I vacuum frequently. The tank and ornaments aren't slimey. It isn't green. The filter gets cleaned about every 2 weeks-changed if needed at 3-4 weeks. The bio-bag balls stay with the tank water while cleaning so I don't lose any good bacteria.

I even tried some of that chemical water clearer stuff!

What the heck am I doing wrong?!!!

Do I just have bad tap water? Is it the decorations? Lack of gravel on the bottom? Do I need a bigger filter?

Arrrrrgh!!!!!!!     Helllllllp!!!!!

I don't know the water is like looking through a murky pond-just never clear. I tried another chemical clearer from API and changed the carbon. It seemed to help somewhat. I don't want to keep disturbing the water too much.

It's a 25 gal tank. 3 fish and the filter is a AquaClear 30 supposed to work for up to 30 gal tanks.
I could go up to a 50 they have them on sale now.
I forget what the water rate turnover for it is without finding the instruction sheet for it.

I tried taking the decorations out for a bit to see if it helped but I felt sorry for the fish. A day later they were looking for their cave. They love to hang out inside it especially the black moor. He goes in there to get away from the other two who fight at feeding time.

I'm going to test the water later and see if the numbers are telling me anything weird.

I think you may be over cleaning. I know alot of people don't have gravel in their tanks and I'm not againest not having it, but don't alot of beneficial bacteria live in the gravel so you not having gravel may not help? I would just leave it for a bit, just do a water change once a week and keep up the beneficial bacteria. It should sort itself out and if your fish are fine and the actual water quality is fine then I wouldn't worry about it.

If you don't have gravel, the bacteria don't have anywhere to settle, except the filter, and it sounds like it is a rather small filter actually, for keeping goldfish, and especially three of them in a 25 gal.
I'm pretty curious about how long this tank has been setup, and also whether you use any water conditioners.
If you're feeding flakes, they could easily cloud the water if you're using some bad quality flake, so if I was you, I'd skip the flakes completely, and buy some nice pellets for your fish instead. They're easier to eat, and more suitable for goldfish over 2 inches.