Managing water in a small pool is a lot like managing water in a big pool, except you get the luxury of being able to easily dump it and start over. Having said that in additoin to bleach you need 2 things, first you need to get some CYA stabilizer in your water (this is available at wal-mart, lowes, etc.) from memory 4 pounds probably cost around $15. Pool calculator says a 10x6x2 ft pool holds 900 gallons, you will want about 30 ppm of CYA in your water to buffer the chlorine from sunlight (too much CYA is also bad), for your size pool you want to add 3.5 oz by weight of 3.8 oz by volume (say a touch under 4 oz on a measuring scoop, you need to be close, but don't have to be perfect) of CYA stabilizer to your water. CYA disolves very slowly in water, the best way to get it is to tie it up in an old sock and kneed it several times a day until it is gone.
You will also need a way to test pH and Chlorine, I suggest buying the cheap $10-$15 drop based OTO Chlorine / pH test kit available at Wal-Mart, Lowes, Ace, etc. (you don't want test strips, they may look easy, but are notoriously inaccurate), while your there you may want to buy some pH down (dry acid). With the test kit you can maintain safe chlorine levels in the pool, once you add the CYA stabilizer, try to maintain 3-4 ppm of chlorine, don't let it drop below 2 or get much over 5ppm (the OTO test will only read up to 5 ppm), using the pH test match the color on the pH scale, you want to keep it between 7.4 and 7.8, use the dry acid (pH down from the pool department/store) to lower pH (a little goes a long way, you wll likley be working with 1-2 ounces at a time) and use Soda Ash (Arm and Hammer Washing Soda) (or pH up which is the same stuff, just cost more from the pool store) in the unlikely event you need to raise pH.
Test daily, dose as needed