Thursday on the way back from the mechanic the battery not charging message and red light came on and I'm not sure if it's an alternator or battery issue. I charged the battery over night and in the morning it read 13.66 volts on the multimeter. I left the car that day and over night with no charger and the next morning it was at 12.55 volts. I started the car and with everything on (heater, all lights, stereo, etc) the volts dropped to 11.75 volts and with everything off the voltage went up to 13.80 volts. I took the car for a brief drive with everything on and the message and red light returned and when I turned the car off the multimeter read 12.22 volts. Does this sound like a battery or alternator issue?

 

With the car running, what is the voltage? If it is not just north of 14v you have a bad alternator. If it is above 14v running, and you have these symptoms, you have a bad battery.

 

This part of your statement is wrong " If it is above 14v running, and you have these symptoms, you have a bad battery. " if its 15 or 16v the regulator in the alternator is bad.
If alternator is 14v ish at a fast idle with no loads and the WL is out the alternator is OK