Time

I am going to restart Saltwater to be rid of all the things I did wrong. Is this a good idea? How do I ensure that green hair algae won't come back and how do I prevent red slime algae? What do I need?

It is a 20 gallon tank, I have two power filters, one replaces the power head I lost. That is all the outside filtration my tank gets.

I know I need a skimmer but is that all?

ATP

For your 20 gallon..

get a protein skimmer (HOB) unless you want to make like a 15 gallon sump or another 20 gal tank and put in sections that's made out of acrylic and you could make a refugim.

As i said before, the main thing that causes your red slime problem is that you're using tap water. Even if you have a skimmer I bet the red slime will grow again if you keep using tap water.

You could either buy a RO unit which will save you money in the long run or you could buy a container of RO at your LFS.

The other thing is, don't use to much daylight lamps. You need more actinic. Even it out.

You could even put some macro algae in your tank if you don't have a refugim. Just stick it to your rocks.

Also, if you're using the same rocks, I recommend you kill the rock (by letting it dry out on the sun for a day or two) and buy a little bit of LR and cycle them together.

Also, make sure you add things accordingly. invertebrates, fish, corals. little by little every week. (when you see algae). be patient.