2002 Mitsubishi Lancer Battery and Idle Problems

My car battery has been acting weird lately. I will charge the battery one night and the next day it will seem to be almost dead after a little bit of daily driving. When it dies, I wait a few minutes and the engine will start generally. This is followed by my car idling at 2500-3000 rpm for 5-10 minutes. The car battery will then be good for another day then will need charged the next night. Today, the car's idle speed did not drop down from the 2500-3000 for 30 min, even while driving. While driving, the engine rpm would surge when I was below 2000 rpm. I had a thought that the car is idling high in order to make the alterratic belt spin faster causing it to charge my car battery more it if was low. My low alternator output lights have not come on and I replaced with alternator with an OEM 80 amp 6 months ago. What should I look at Or replace?