According to this catalog http://www.bikeman.com/attic/catalog...992catalog.pdf your bike has Shimano Rapid Fire trigger shifters. These type of shifters have a ratchet mechanism which allows you to pull cable to shift to larger chainrings (front) or cogs (rear) one or more clicks at a time and an escapement mechanism which allows you to release cable only one click at a time, when it is operating properly. It sounds like one of the escapement pawls in the shifter is either broken or sticking, possibly due to hardened grease in the shifter and is allowing all of the cable to be released at once. There should be one or two small screws which hold a cover on the shifter, I'd suggest that you try to open it up and have a look inside. The mechanisms are usually pretty easy to figure out and some cleaning and re-lubrication may set it right. If there is a broken part you should find it inside. Otherwise there are replacements available from Amazon and EBay. I can't tell from the catalog whether the shifter is separate or integrated with the brake lever.
Example for separate shifter: http://www.amazon.com/Shimano-Acera-...s=SL-M310+left
Integrated shifter-brake: http://www.amazon.com/Shimano-Acera-...mano+EF51+left

Edit: Pairs are available if having mis-matched shifters is too OCD-inducing.

Some clear, well-lit and focused photos of your shifters may help us to give better edvice.