Hi Chris.
As you're new to the hobby I'm going to, very simply describe the basics of fishkeeping, which no one seems to have done yet! I'll keep it simple (don't take it as patronising in any way!) as I know how confusing it can be at the beginning.
Fish produce ammonia as a waste product. The ammonia is very toxic. Luckily there is a family of bacteria that eat the ammonia and turn it into nitrite. Nitrite is also toxic to fish.
There is another family of bacteria that eat nitrite and turn it into nitrAte, which is only toxic to fish at very very high levels. We keep the nitrate in check with regular, partial water changes.
Growing a colony of those good bacteria in the filter is what we call 'cycling'. Once you have the bacteria, your tank (well, technically your filter) is 'cycled'.
As there are no fish living in your water supply, there shouldn't be any ammonia or nitrite in your tap water. So you take some of the water from the tank out and replace it with new water from the tap to make sure the fish aren't being poisoned.
As your tank is already established from before you had it, it should already be cycled, but sometimes the bacteria don't like being moved and the colony dies back a bit. This is probably why you have a reading for nitrite in the tank. You need to keep doing water changes, large enough to reduce the nitrite to zero, until the bacteria recover.
You might also have ammonia present so, until you can get a test kit for that, I's suggest you do a 50% water change every day.
The bacteria do cling quite strongly onto the filter media (that's all the stuff inside your filter), so hopefully you haven't washed too many away, but in future be a bit careful cleaning the filter. You want to clean it quite gently so you get rid of the clumps of dirt and poo, but not so much you dislodge the bacteria.
The way you've done your water change is perfectly correct, though if you're treating water in the bucket, you only need to treat for that volume, not the whole tank.
Now lets move on to your fish.
The black one with the red fins is a rainbow shark.
In the picture you've posted with three fish in it;
the one at the back that is blue/grey with vertical stripes is, I'm 99% certain, a convict cichlid
the one in the middle with the spots and the yellow end to it's tail is a green terror
The silver/gold one at the very front is an arulius/filament barb
I'm afraid to say, your tank really isn't big enough for those fish. Keeping fish in too small a tank causes them to be stunted, and not grow properly. This significantly shortens their lifespan. It may already be too late for those fish, if they've been in there three years, but they might well grow to nearer their proper size if you could get them a bigger tank.
Now you know what species they are, you can read up on them yourself and make a decision as to what you're going to do.
Hope that helps you with some of the basics; best of luck!