ACK! I really don't know what to do and I'm very concerned for my goldfish...
I have a 37 gallon tank with two sarasa goldfish, one shubunkin, two cory catfish, two pictus catfish and a plecostomous. The pictus cats are a fairly recent addition.

Here's my issue- a few days ago I noticed one of my sarasas had what looked like the beginning stages of Ich on his tail and a few faint, thin red veins in his tail and a small tear in his tail fin. I figured I would wait a couple days just to make sure it was indeed Ich. Then a day later I noticed a redish, spot on his body just before the bottom of his tail. However now all THREE of my goldfish have tears in their fins with thin red veins running to those spots with FAINT little white spots that I feel like were not there before... All three of them spend most of their time at the bottom of the tank not moving at all now with the exception of swimming madly around the tank and splashing about near the top. Fortunately they do still seem interested in eating...

A about 2 months ago I had a butterfly that started exhibiting the EXACT same issues. Tears in his tins, little red blotches, hiding/lethargy, but occasionally swimming erratically and splashing near the surface. Unfortunately I could not figure out what was wrong so I separate him and dosed him with melafix until I could figure out the real issue. I had to move and give him to a good friend before I could diagnose him, but he's still alive and healthy according to his new owner!

PLEASE PLEASE help me out folks! Thank you!!!!

Great, I really appreciate your response!
I'll do what I can to snag another tank to move everyone else into... I tested the water recently and the only thing that was high was the ammonia, of course, from the goldfish. So I'll do what I can to lower that.

My one concern now is that the koi I had that showed these same symptoms continued to show them for an extended period of time even after I removed him from the tank and put him in a 30 gallon container all on his own... Can you think of any actual diseases that might cause this instead?

I'm pretty certain it's simply water quality issues, you'd better change some big amounts of water for a few days while testing regularly. If you want to, you could add some salt and melafix to the aquarium to help them recover.

And also, give away some of your fish, you can't keep any more than three goldfish in that tank.

I am glad to hear that, you are doing the right thing.

But can you please let me know how high the Ammonia readings are?
You will also need to test for Nitrite and Nitrate. As you know Ammonia is broken down to Nitrite which is still very toxic to fish and then Nitrite is broken down to Nitrate. Nitrate in higher concentration is toxic to fish too.

As real plants consume Nitrate as fertilizer it is important to have those in the tank too. They also produce Oxygen for the fishes to breath, and keep the algae growth down while using up Nitrates.

What sort of equipment do you have? Filter, airstone?

Can you please post the numeric results of your water tests?

How do the fishies behave now?

Maybe you can use this filter in your new tank and get a bigger one for the goldies tank? It should turn 2-3 times more water than the tank volume is per hour.
I know they recommend filters for a certain tank size: expweerience showed though that it is much better to have at least 2 sizes up compared to what they recommend.... much easier maintenance.

The fins will take a while to recover, use Melafix, this will help.
And yes get yourself that missing test kit asap.

Plants: I use the following: Ambulia, Ludwigia, Wisteria, Elodea, Java Fern, Hornworth, Anubias, Ricchia, Fontinalis, Amazon Sword Plant. The Anubias (dwarf one) is planted to a teracotta vase in the goldies tank and in my tropical tanks on pieces of drift wood.

Nitrate isn't harmful until it reaches 40ppm, but it's ALWAYS better to keep it somewhere below 20 if possible. And you said "testers"? Does that mean that you've got paper strip tests? In that case, buy a proper test kit with bottles of reagent instead, because the paper strips are usually quite inaccurate, and the bottles last for ages!

Now, you definitely need to get a bigger filter, go for an external power filter, as the internal ones are usually too small to handle the waste of goldfish. Ideally you should get a filter that turns 5 times the tank volume at once, and it should rather be rated for a tank twice as big as yours. I've got an external power filter myself, about 1000L/h and an internal one (smaller) turning 1500L/h, so I've got about 7 times the tank volume being filtered per hour. The small filter is rated for a 260L and the external one for 600L. So it's good to combine two filters as well In fact, the more the better, "overkill" is always to prefer when it comes to filtration, even 10 times the tank volume would be nice.