A brief note on technique.
A razor is, in effect, a line. Possibly several parallel lines forming a plane, but when you get down to it, a line. We also, for simplicity, approximate a face and neck as a cube.
Yeah, right.
Your face is not a cube. It's a collection of compound curved surfaces. To make matters worse, beards have a distinct grain - a direction the hairs lie - and the grain isn't constant at all. For example, on my neck the grain tends to run right to left, except for a patch the size of a quarter about halfway along. And I'm probably relatively normal.
So good luck running your line over your collection of curves.
There are better shavers than me out there (quite a few of them, in fact) and they all recommend the same thing: multiple passes and short, light strokes. I follow that advice to a greater and lesser extent, but again - I've got an easy case. My "beard" (such as it is) is pretty light and not very wiry. Some others who shall remain nameless (*cough* *cough* B... *cough*) apparently have facial hair that could serve as a wire bristle brush.
So, the recommendation from the professionals:
Three passes per area. First with the grain to get most of the hair off, then across the grain (at 90 degrees) to get closer, and finally against the grain for that "smooth shave" that women seem to be so attracted to (as a note: they're not, but I can keep dreaming). Short strokes of about an inch to minimize the chances of nicking.
I almost hold to this. There are certain areas (sideburns, for example) that get away with a single pass. My chin, especially the underside, gets three or four. I end up doing a sort of rough shave then using my hand to find patches I missed. Flat bits (sideburns, neck, etc.) get a single sweep; curvy parts get short strokes.
Lastly, and possibly most important: keep your face wet as possible at all times. You should have a brush that is pretty well lathered, so use it. Dry shaving is... uncomfortable. I tend to splash my face wet, then apply lather - and take your time, you owe it to yourself. Lather up, do one set of passes with the razor (rinsing as appropriate, of course). Then check for missed spots, splash your face again, and lather again. When I've gotten impatient and said "eh, I don't need to brush my face more"... there go the nicks. And cuts. And slices. And OH MY GOD THE BLOO... just kidding. Mostly just nicks.
Oh, and when I'm done, I put on some facial moisturizer. I've found that the razor manages to take a lot of the surface oils off, and besides - winter in WIsconsin. I'll take anything I can get.
A word about bleeding. You will. It depends on the razor blade (yes, there are different types - I'll get to that tomorrow or sometime), but chances are you'll nick yourself. I don't worry about it any more. The small cuts seal over nicely with the moisturizer and bigger ones get handled with a dab of toilet paper. Besides, with a sharp blade, they don't hurt anyway.