Is my battery bad or alternator?

Just about any place that sells car batteries will happily test your charging system for free. 12.5V is a bit low for a battery that has set overnight (12.6 to 12.7 is normal), but in the northern hemisphere the weather is getting cold so voltage will be down a bit anyway.

If you want to check it out yourself, my favorite YouTube mechanic, ChrisFix, has the following video. The one thing I would add is to measure the AC voltage across the battery. If the AC voltage is above 0.1V (usually 0.4 or more) the alternator has bad diodes, which limits its output current.

https://www.youtube.com/watch?v=LGB6ZEjGm7Q

12.5 volts with the engine running? Bad alternator. 12.5 volts with the engine off? Normal. You need to take it in to have the system checked. Make sure you aren't leaving the interior courtesy light on.

Since it would seem you have a volt meter how about hooking it up to the battery posts and with the motor running and at 1200 rpms check to see if you have 13.7 volts or more. If you have less than 13.7 volts there is a bad diode in the alternator and you are not charging the battery. If you've been jump starting the vehicle you should check the fusible link for the electrical system to see if it got blown. The alternator can't charge the battery if the fusible link is blown. The fusible links are the bigger fuses in the engine compartment fuse box.

That video is complete nonsense. Measuring from the negative terminal to the alternator case doesn't tell you anything. Just measure the battery with the engine running and the headlights on to check the alternator.

Go by a auto parts and have them test the alt. and see if its working correctly. They do that for free.

You can test both with a multimeter. Google it.

I think you should sell that piece of **** and buy a trans am.