91 Tercel Battery light question

I was driving the other day and the battery and brake warning lights came on. It is a manual and when I would rev the engine, the lights would go off. As it idled lower, they would come on again. After a while, they came on and stayed on. If I let it sit over night and then turn it on and drive it the next day, the lights won't come on for 10 minutes or so. I started it this morning, and while it was running, I disconnected the positive from the battery. The car conitnued to run. I could put the fan on low, med, or high and it would still run. If I turned on the lights, it would cut out. Would this seem to be the battery insead of the alternator? I'd hate to go buy a battery if the alternator was the cause. I checked the voltage before starting the car and it was right around 12v. With the car running it varied between 13 and 14v.

Any advice? A couple guys at work said to check the wires on the alternator. I haven't done that yet. What should I check and what should I expect? I'm by no means a car guy but if it's something simple, I'd like to save the money. How hard is it to replace an alternator? The space is pretty tight and I don't have access to a shop. I'd need a step by step to change it as I can't find my repair manual at the moment. Thanks.