The old battery was over 4 years old and was starting to go. When the phone got below 20% it would frequently shut down until it was charged again, and was quick to drain.
The repair went well. I used the suction handle tool to open the phone, although it opened a little too fast and pulled out the home button connector (and scared me for a second) but fortunately there was no damage done to the home button. I held the screen up using a can and rubber band so that I didn't need to disconnect it. I used a hot air gun on the back of the phone for about a minute to loosen the battery adhesive, which worked great for one of the strips (got it completely off with no issues) but the other strip snapped as soon as I started pulling it. With one strip off I was able to easily pry the battery off of the other adhesive strip using the flat end of the spudger tool. After that, installing the new battery was easy, and I connected everything back up and put the phone back together. The phone booted back up with no issues, and I let the battery drain below 10% before charging it fully as instructed for the new battery calibration.
I should have followed this tip but didn't. Put some tape on the phone (on the screen wrapped around the back) so that when pulling the phone open, the tape stops it before the home button connector gets pulled. Luckily my home button wasn't damaged and I was able to reconnect it normally, but you risk breaking it doing it the way I did (since I only had the suction handle and not the iSlack).
Also, definitely use something to heat up the battery adhesives before pulling them, and make sure you pull them slowly and carefully so that you don't break them. I was lucky that I only broke one of them.