How to change the aperture on my Nikon D40?

I set my camera to the A setting, and I was using the command wheel (the one on the back, not the top) to change the numbers when i looked through my viewfinder. The numbers are changing, but I don't see any change in the depth of field. I put my hand in front of the lens, but far enough for it to focus, and my hand and the background were both equally as focused. What am I doing wrong?

Oh, one more thing: how do aperture, shutter speed, and iso play into each other? How should I adjust them to compliment each other?

You will not see any changes in the field of view even when changing aperture. Lenses stay at their widest open value until the shot it taken… THEN it closes down to whatever aperture you have set. This is done because if the lens actually went to say f11 before taking the photo, your view through the viewfinder would be extremely dark. Also, the auto focus system would not be able to function at the smaller aperture.

Some cameras (I do not know if the D40 has this or not) have a DOF Preview button. You push that button and it stops the lens down to your set aperture before the photo is taken. It will show the depth of field as it will be in the shot, but in most cases, the view becomes so dark that you can't really judge anything about it. The Depth of Field Preview button is truly a worthless item to me, and many cameras no longer even have it.

Here is a link to help you understand the relationship between ISO, shutter, and aperture:

http://teenagephotographer.com/the-basics/

In the early days of 35mm SLR cameras there was no communication between lens and camera. You composed your picture with the lens wide open (at its largest aperture, i.e., f/1.4 or f/2 or f/2.8) and then stopped down to the aperture you wanted to use to actually take your picture with, metered, set the correct shutter speed and released the shutter. Then in the early 1960s camera manufacturers invented a way for the camera and lens to actually communicate. Now the lens would remain wide open regardless of the f-stop you wanted to use to take the picture. So if you wanted to use f/16 as your taking aperture you selected f/16 and circuitry in the metering system would adjust the sensitivity of the meter to measure the light as though the lens was at f/16 to give the correct exposure at f/16 while the lens remained at f/1.4 or f/2.0 or f/2.8. Once you set the correct shutter speed, as you pressed the shutter release the lens stopped down to f/16 This is called "Meter Coupled" and is what you're experiencing with your D40.

NOTE:
Although some cameras have a DOF Preview button (your D40 doesn't) over the last 43 years I've found it to be all but useless even on a sunny day.