No need to go to the hardware store for a kit. The best kit and the one that we recommend is the Taylor K-2006 or 2006C which you can get through the test kit link in my signature below. Also, in my signature, please read the Best Guess Chlorine Chart that shows you the needed chlorine levels that correlate with a pool's CYA level.

I am assuming that this is not a liner pool. Correct?

Your pH is too high. You need to use some muriatic acid to lower it down to 7.4-7.6 although anywhere 7.2-7.8 is ok. There is a guide to using muriatic acid safely in my signature also. You'll want to adjust your pH before shocking the pool since high chlorine levels make the pH test inaccurate.

TA is fine and your CH is ok. I would not advise using any calhypo, though. For a gunite pool, a CH reading of 200-400 is the goal. You're a little past that now, but ok.

Go ahead and shock the pool up to around 15ppm. In your pool, one gallon of 20% sodium hypochlorite will add about 10ppm of chlorine. Use that as a reference to help you to figure out how much bleach to add each evening when you test. (Again, order a good kit ASAP.)

If you can, scrape a little of the suspected black algae up with your fingernail and scrape it on a white index card. If it smears a dark green, then it probably is black algae. (So, why don't they call it green algae then, right??? )

Hope this helps!