How to shave
It always pays to prep your face with a hand towel which has been soaked in hot water. Give yourself two 30 second increments, using water as hot as you can comfortably stand. There's no reason to scald yourself. In a hurry, fresh from the shower will do. The point is to get those whiskers soft before you reduce them.
Lather your face
Please don't buy a fancy razor and pair it with goop from a can. Nothing can surpass a high quality shaving soap or cream which has had air and water whipped into it with a good quality shaving brush. The "best badger grade" is a good choice; "pure badger" brushes tend to be of lower quality. The choice between soap and cream is a personal one. Cream lathers a bit more easily and, containing glycerin, is more protective. Soap gives you improved feedback from the blade, and also permits a closer shave.
You can use a shaving mug or lather on your face. Probably the easiest approach is to lather directly on your face. For soap, swirl the puck with a just damp shaving brush until it is well loaded, then add a small amount of additional water directly to the brush. For cream, if it's spreadable enough, you can put some on the brush with your finger. If not, spread a thin film onto your face, then apply the brush. Use gentle circles at first, followed by a back and forth "painting" action. Many people don't use enough water. Don't worry if the lather seems thinner than you are used to. If it is too frothy and forms peaks which are too stiff, the mix is too dry. If it is too thin or has not been whipped enough, it will appear bubbly and runny. You are looking for what cooks call soft peaks; you should have a coating of lather with some body that lubricates your face and over which the razor glides easily.
With experience double edge shaving, you will get to know every whorl, every spot along your jawline where stubble hides, more intimately than you know your partner. The technique takes some time to master, but it turns shaving from a chore into something to look forward to.
In double edge shaving, the beard is reduced progressively, using a succession of three passes: with the grain, across the grain, and against the grain. Don't try removing all your stubble in a single pass. Though it is possible to get a quick shave in single pass, you will inevitably leave some stubble behind. If you have fifteen minutes to shave, use three passes.
The first pass is in the direction of grain growth. This means in a generally downward direction. There should be very little friction. You can feel and hear the blade cutting. After one pass, the shave will get you through to five o'clock, but not much better. It's okay to go over a troublesome area more than once, especially if you rewet it slightly and "borrow" a bit of lather from another part of your face. But be careful with this technique. Too much shaving of a single spot produces skin irritation. Moreover, you will catch those trouble spots on subsequent passes. Rinse off, but not too thoroughly. The idea is more to rewet than it is to rinse.
The second pass is across the grain growth. The direction of the shave means different things to different people, depending upon growth patterns. In general, you will work from side to side, shaving to the inside. Again, the razor should glide smoothly. You should both feel and hear it cut, but there should be no sensation of skin irritation. You are prepping here for the third pass, so you have to get pretty close. I shave first inward, then outward, using the rewet and borrow lather technique. When the second pass is completed, your stubble should be pretty flat.
The third pass is the money shot, the pass that separates the men from the boys. Now we are going against the grain. The shave is generally in an up and out direction, feeling the grain of the beard in feedback from the razor. Sometimes it pays off not to go too hard against the direction of grain growth, especially if you feel incipient irritation. Other times, you can let 'er rip. Your shave will be sensitive to beard prep on a given day, the quality of prior passes, even ambient temperature and humidity. If you can manage to cut directly against the direction of grain growth, you will be rewarded with the smoothest shaves of your life. Rinse down thoroughly, and towel off. You should be as smooth as a baby's bottom, with no razor burn. Any minor redness should dissipate in a few minutes, or you've shaved too hard.