My battery was definitely showing its age. Full charge capacity was only 702 mAh even though the cycle count was 438 (less than 500).
After reading all the instructions and lots of comments, I was a bit apprehensive about this but figured the worst that could happen is that I have to buy a new iPhone. I'd rather wait for the next version. This iPhone 6 does everything and more than I need. So, I prepared my workspace (kitchen counter) and started. Thanks to the many comments and the instructions from iFixit, I went super slow on pulling the adhesive tapes out and it worked perfectly. I dry fitted the new battery in place before putting on the new adhesive strips, so there was no problem attaching it. I did not remove the display but instead propped it open - used the suggestion to put a rubber band around it and a box. If the dogs hadn't need to go out in the midst of my repair, I would have finished faster.
When pulling the adhesive strips out, it seemed at first that they weren't going to budge but I just waited while tugging slightly and they began to move. If you have it completely quiet in your work area, once they start to move, you can hear them popping as they let go. It still took longer than I thought it would to pull free, but they both came out completely clean.
The advice from others to dry fit the new battery was a really good idea. The instructions to place it in on the side edge and then flat are also a good tip.
I used some tape on the counter to hold the screws while I worked.
As someone else mentioned, when you start to take the tab of the adhesive off the old battery, be sure you're getting the white adhesive under the black since it's the white adhesive you need to pull out.