Lift the battery as a whole up out of the upper case, and remove the battery.

With a little luck, you can slowly pull out each strip of adhesive with your fingers.

Otherwise, soak each section of adhesive with a bit of adhesive remover for 2-3 minutes, and then scrape it out with an opening pick or one of the other tools in your kit. This can take quite a bit of work, so be patient.

Mop up any remaining adhesive remover and give your MacBook Pro a few minutes to air dry.

Calibrate your new battery before using it: allow it to drain overnight, then charge it to 100% and drain it again until your MacBook Pro shuts down automatically. Charge it again and use it normally.