05 Golf dies after startup

I just bought this car, it is a auto with 137000 on it. The first time I started it up cold I put it in reverse and it wouldn't move. I waited 5 minutes with it running and then it took off. It did this the next time too, then it started dying immediately after I let off the starter. It starts back up and I let it warm up and there are no more issues. This is my first PD. I am going to change the fuel filter today, if that doesn't fix it could it be the lift pump?

Check your fuel filter; it should be full. If not then yes your in-tank pump is not working. You should be able to hear it for two seconds with the ignition turned on and the passenger seat lifted.

Most likely the lift pump. When you've got the filter off cycle the key, you should get about 50cc's of diesel from the feed line from the tank. If you don't, you've found your problem.