By reading everything on here, I took my 15' Intex AGP from a festering green, dirty mess into a crystal clear, sparkling pool in about a week of fairly intensive study/work. Before that I was at a loss about what to do and how to do it.
First I upgraded to the 2650 gph Intex sand filter. This was important both as a greatly improved filter and also because it gave me the first chance to properly vacuum all the junk off the floor of the pool which I had struggled to remove through many harebrained schemes, none of which was adequate.
Then I got to work with the shocking process, skipped the algaecide and phosfree stuff the pool store tried to sell me on, and finally got the pool holding free chlorine after a couple days of continually keeping the FC up at desired shock levels with CYA at 30. This took a lot of chlorine, but it was the turning point.
This morning, after a good vacuum last night, and the filter running 24/7 for about the 3rd straight day, I woke up this morning to a gorgeous crystal clear pool.
I finally feel in control of my pool, whereas before I just felt a sense of despair that I didn't know how to go about keeping it clean.
Yes, I dropped about 3x as much as I originally had planned to spend on the upgraded filter, the good test kit, etc., but it's still a dirt cheap bargain compared to an inground pool, and there's no point in saving money on a pool you don't want your kids to swim in.
A big thanks to all who volunteer their time and effort helping the new owners out here.