What is involved in DIY swimming pool maintenance?
In my experience, most horror stories of pool maintenance are usually the result of neglect, or not paying attention to the instructions.  It's not that hard, but expect to spend an average of 10-15min per day checking on your pool, adding chlorine, clearing out leaves in the skimmer, etc.  Also, once a week, expect to spend some extra time brushing/vacuuming the pool and cleaning the filter.

As a first time pool owner, the simplest thing you can do is pick up a free pamphlet on pool care.  Follow the instructions in the pamphlet, and make sure you stay on top of it.  My Wal-Mart usually has pamphlets by HTH Pools next to their chemicals.    

Start by getting a test strip kit.  You dip the strip in the water and compare the colored pads against a chart.  Pretty easy (unless you get into a fight with your wife about the shade of orange of the pH pad).

Make sure you pick a good test kit, which measures several properties of the water (should measure pH, stabilizer/CYA, alkalinity, hardness, chlorine).  Cheaper ones will only measure a few properties.

Measure your pool every day or so.  If the levels are off, adjust them by adding the appropriate chemicals.  Just follow the instructions on the container.  Most any chemical you need can be found at your local pool store, or Wal-Mart.  Beyond that, I'm not going to repeat what you would find in the pamphlet. (Did you get one yet?)

One bit of advice that wasn't in a pamphlet:  If you buy "stabilized" chlorine products, they will raise the level of stabilizer/CYA in the pool.  If your CYA levels are already high enough, the stabilized product can push your levels over the top.  If you aren't sure if the product is stabilized, look for the words Dichlor or Trichlor in the ingredients.  Un-stabilized chlorine is usually Calcium Hypochlorite or Sodium Hypochlorite.

Beyond the chemicals, there will be some "mechanical" maintenance/care:


Dirt, pollen, and algae will naturally collect in all the crevices  of your pool, so use a pool brush to break it up.  Regular churning of the water (from brushing or swimming) will give the filter an opportunity to suck up the debris before it settles.  
You will need a vacuum (manual or automatic) to suck up non-floating debris.  48 hours before vacuuming, shock the pool to make sure any algae is dead.  24 hours before vacuuming, add some clarifier to make the algae and dirt settle on the bottom (where the vac will get it).
Make sure your filter is clean.  There should be a pressure gauge on the filter that will indicate when too much crud has built up in it (pressure levels will go up).  At that point, you need to clean it.  Some filters are cleaned by backwashing.  Others have a cartridge that needs to be changed.  The filter should come with some instructions on how to care for it.
Check the strainer baskets in your skimmer and pool pump.  Larger debris like leaves and large bugs will collect there.  Periodically shut off your pump and scoop the crap out.
A plugged filter or strainer basket will strain your pump, and maybe cause a burst hose (very bad).  So, keep the water path clear.
If you have cold winters, make sure you follow the instructions on winterizing the equipment.  I disconnect my pump and bring it indoors every year.