Hi, and welcome, Cloud.
Bleach IS chlorine, and no, once in the water it doesn't "battle" other forms of chlorination.
In the water all chlorine is chlorine. But different forms have side effects, NONE of which have anything to do with chlorine itself.
So I'm not sure what the WalMart "Shock" is, but I'm going to guess it's 48% Calcium Hypochlorite--"Cal-Hypo". The other alternative is that it's Di-Chlor.
Here's a quickie rundown on the common Chlorine forms. Once in the water, the chlorine they deliver is all the same.
1) Tri-Chlor. Typically 3" or 1" tablets that dissolve slowly and go in floaters or in-line chlorinators. In addition to chlorine, they are VERY acidic, lower pH a lot and are, by definition, "Stabilized", meaning they add Cyanuric Acid, aka, Stabilizer, Conditioner, Balancer or just CYA to your water. Occasionally sold as a powder.
2) Di-Chlor. A powdered form similar to Tri-Chlor. It is not acidic BUT for various reasons it will lower your pH, but not as much as Tri-Chlor. It is also "stabilized" and adds CYA to your water in large amounts as well.
3) Cal-Hypo. Mainly sold in powdered for as the tablets have myriad problems. It can be in 48%, 62%, 68% or even 72% strength. Frequently sold as "Shock". Cal-Hypo is a non-stabilized form of chlorine, so it adds no CYA to your water. It usually raises pH somewhat, but adds calcium into your water. Since you have a vinyl pool, calcium is useless, but harmless unless it gets too high. Then it can cause scaling on metal and make your water go milky.
4) Bleach. Sold in pool stores as "Liquid Chlorine" or "Liquid Shock". Ordinary bleach is 5.25% concentrate, and Ultra is 6%. Pool store Liquid Chlorine (LC)can be 6%, in which case it is IDENTICAL to ultra bleach, 10%, or 12.5%. The strong stuff, 12.5%, is LC if it's sold in blue 5 gallon drums, "Liquid Shock" if it's sold in gallon jugs. It's all Sodium Hypochlorite, a chlorine concentrate. Bleach/LC adds nothing to your water besides chlorine (maybe a little salt). It doesn't drive pH up or down, doesn't add stabilizer or calcium and it gets into the water FAST! It's biggest drawback is its bulk.
5) other less used or commercial chlorines: Lithium Hypochlorite. This stuff is worth its weight in gold. It must because it COSTS its weight in gold! Actually, it's an expensive waste of money. Gas chlorine. This MUST be done by professionals and is usually used in large pools. I know little about it.
I suggest you start by going to the sister site, PoolSolutions.com and reading the tips and articles there. Pretty much everything we'll tell you here is there in easy-to-read and find formats.
Then start reading the stickied posts here in the various forums.
One last note. The Intex SWG is probably too small for your pool. I think it's max is 11,000 gallons and at that it's going to work hard and probably not last too many seasons. BTW, Salt Generation is NOT an alternative to chlorine. It IS chlorine and is totally compatible with the other forms above.
Carl
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