i have had a few cheap backyard frame pools, and pretty much the "filtration systems" they come with are only good for people that never get in there pool dirty, have no friends, no kids, no dogs, nothing but time to wash or replace the filter constantly, always keep the pool covered, you get the idea.
my pool is a 16' round 42" deep. think it holds about 4000 gallons. basicly between all the people, and our 2 filthy dogs, the little cartridge filter was not cutting it. it hangs in there for the 1st couple weeks then the water just gets too nasty. even though i come home and sray it off every day and change it once and a while. the filtration setup that comes on the pool should probly only come on somthing with a few hundred gallons. whatever, i think it sucks. in the picture obviously im also dealing with some algae.
Step 1: the basics
why do you want a sand filter? because sand is actually a really good filter, thats why most larger pools and aquatic facilitys and even your nicer backyard above ground pools use them.
how do they work? well its actually really simple. a "normal" sand filter is a sealed container half full of sand, water is forced though the sand by the pump. ya its that simple.
how exactly? a pump forces the water into the top of the container, usually through somthing that spreads it even so its not just a jet of water blasting a tunnel in the sand, you want a nice even sand bed the water has to go through. in the bottom there is somthing that usually looks like a a bunch of little pipes with slits in them the sand cant fit through that connect to a big pipe, that is the outlet where clean water comes out.