I just spent (and wasted) a LOT of time diagnosing electrical problems on my roommate's 95 Mercury Mystique (Mistake), and by not following the proper steps, myself and the other person working on it wasted a lot of time trying to solve a simple problem.
The first thing you need to do is get or borrow a voltmeter. You need to check the voltage at the battery and at the alternator both with the car off and with the car running (4 voltage checks). Let us know what they are, and we can tell you where to go next. When you start making guesses without checking voltages, you might as well throw darts at the car and start replacing parts.