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 newly installed battery: charge it to 100%, and keep charging it for at least 2 more hours. Unplug and use it normally to drain the battery. When you see the low battery warning, save your work, and keep your laptop on until it goes to sleep due to low battery. Wait at least 5 hours, then charge your laptop uninterrupted to 100%.