Hi, and welcome to the forum!
First of all, if you're trying to manage your water without good testing, it's a crapshoot--you need to invest in a good, drop-based kit. WalMart has a $15 5-way that will be a good one to start with.
Second, slippery skin often is a sign of either gunk in the pool that needs to be destroyed, or possibly an alkaline situation. Since you have no idea what your pH is, or your alk, it's going to be impossible to give you good advice. If you will please run a set of drop-based results for Cl, pH, TA, and CYA and post those numbers here, we'll be glad to get you up and running.
If you're having trouble holding a chlorine level in the pool, then something is burning it up--either the gunk that's making your skin feel slippery, or the sun, or a combination of the two. If it's gunk in the water, then what you need to do is shock the pool--but how high your Cl needs to go to reach shock level depends on your CYA level, so again you must know that first. If it's sun, then your CYA needs to come up. I have a feeling it's a matter of shocking the pool, but without good test numbers it's going to be hard to tell.
As far as your Alkalinity up and calcium, you've been what we call "pool stored"--alkalinity up is the same thing as Arm and Hammer baking soda, just at an inflated price. And in a vinyl pool, you don't need calcium. Calcium is for concrete pools to keep the water from leaching the calcium out of the concrete walls.
As far as chlorination goes, bleach does not mess up the water. In fact, it's the best form of chlorine you can use, generally, because it's the only form that doesn't add other stuff to the water that can mess up your water balance. You can switch around at will with different forms of chlorine, as long as you don't mix them--but once dissolved in the pool water, chlorine is chlorine.
Janet
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