I worked in a bike shop through college and have ridden/built bikes/worked on bikes my whole life. It is absolutely possible to,swap,shifters, derailleurs, cranks, etc. on bikes. But, getting everything to work together can be tricky. Not every combination works well together. Does your bike have a derailleur or an internal gear hub? From your description it sounds like it might have a 3-speed internal gear hub. Does the shifter click through the gears (indexed shifting) or does it go smoothly and you have to select the gear by feel (friction shifting)?
What you are describing sounds like an older style system. If you can post a picture of the shifter and the derailleur, it might help to identify them, as well as your options for changes.
Sheldon Brown created a great site with lots of technical information, especially for old weird bikes (my favorite bikes!).
There are also two books, zinn and the art of road bike maintenance/zinn and the art of mountain bike maintenance written Leonard Zinn. The newest editions are probably full of all the latest bike tech, but an old edition of the road bike book might cover 3-speed hubs.