Why won't my car start?

I have a 2000 Ford Expedition 5.4 liter engine. I currently live in Wisconsin, and the winters have been hard on my battery. Since it started getting cold this year, I’ve been having trouble getting The Beast to start. Up until a week ago, a simple jump would do the trick. Then, last week, I went to start it and it was giving me the usual trying to turn over but slowly sound that indicates that I would need to jump it.

I attached the jumper cables, and there were all indications that a proper connection was made. When I went to start it, though, nothing happened. Turning the key to the ON position has all the dashboard lights come on, the dashboard goes black when I turn the key, and the lights come back on when it returns to the ON position. There is no sound that it is trying to turn over, though.

Our first step was to remove the battery and have it tested. It came back as a bad battery and we’ve bought a new one and installed it. There is still no indication that it is trying to turn over when the key is turned. What should be my next step in diagnosing the problem? As I work through the various components that could be bad and causing it not to start, how will I know if there are multiple components that have gone bad and not just one?