I'm 16 and this is my first car.

Hey so my name is Brad and I have a 98 a4 1.8t 5spd and my car was running good and then one day my battery started dying. It would start fine and then I would park it overnight and it would be dead in the morning. I figured I had a bad battery so I bought a new one and one week later my new battery was dead. I've done some testing hooking up the voltage meeter to my battery and pulling fuses with no luck. Any ideas as to what could be doing this? Any better ways of testing?

I let my car sit for a few weeks and yesterday I charged the battery and started it but when it started it drops down to like 0 rpms and then jumps back up and is steady like normal. When I pull up to a stop sign it drops down (sounds like the car died) and then jumps up to normal. Any idea what this is? Any help or ideas would be awesome. Thanks, Brad.