It's the stem you want to raise. The headset is the bearing assembly.
Are you positive it's threaded? If so, instead of the steerer tube extending up beyond the head tube, and the stem clamping onto that, with a threaded headset, none of the steerer tube will be visible. The stem is an angled piece, the part of the stem that's visible above the headset and steerer tube will be a single piece, with one bolt at the top and one bolt in front to clamp the handlebars. If it's really threaded, then changing the height is easy. Loosen the bolt at the top of the stem, put it where you want it, re-tighten. Just like seatposts, a stem for a threaded headset will have a maximum height marked; make sure you don't go beyond that. (Sheldon suggests that most complete bikes left the factory with the stem at max height though - so you may still have to replace it - which is really a huge PITA with a threaded headset since you have to take the tape and lever off one side of the bars.)
For comparison, here's a picture of the steerer tube and stem for a threadless headset. (This setup is a little odd, in that the builder has used a seatpost clamp instead of spacers to secure the steerer tube, but there's no difference in the stem.)
Images from, and more discussion at, http://www.sheldonbrown.com/handsup.html