Cateye manuals:
http://www.cateye.com/en/manuals/66
I have a 2000 KHS tandem with Shimano triggershifters and newer bikes with SRAM triggershifts. Generally, one trigger on each side goes up , the other goes down. Yours look different but probably work similarly.
I would suspect that your cables and housing could stand to be replaced and if you do that and then adjust them, all will work.
http://www.parktool.com/repair/byregion.asp?catid=53
Spending $10 on the bike (quite a deal!) allows you to spend whatever it takes to make it right.
What is the frame make of?