How do you change from a quill bicycle stem to a threadless bicycle stem?

There are two options:

1. Get a quill-to-threadless adapter, then slap the desired threadless stem on. This lets you keep your current fork
2. Replace fork and headset with threadless versions, then add the desired threadless stem.

Do note that there are several different dimensions available, so you need to get your measurements right or risk buying the wrong stuff. Stems can be both 1" and 1 1/8" by the steerer tube, and 25.4, 26.0 and 31.8 by the handlebar.

Also note that there's nothing particularly wrong with quill stems, make sure that this is worth the effort.