As stated above try a hard reset by holding both the power and home button simultaneously until you see the apple appear. If that does not help do a backup, import your photos, and restore your device.
If restoring does not help I would wipe the device and reinstall the IOS fresh (then setup as a new phone). Do not restore from a backup at this point.
I have noticed my 6 plus does not transition as smoothly with IOS 9.0.2 than my 6S. So I'm actually going to wipe my 6 plus and start fresh.
I've noticed the same thing. My fingerprint doesn't work S well, serious lagging , freezing on certain apps, I just did a hard reset. If it continues I'll do s restore later. I'm on a 6 plus but I also have a 6 and although the 6 has issues as well, not as bad.
Terrible freezing and it's weird for me bc I'm not used to my iPhone freezing pretty much at all - it kind of reminds me of my old androids.
At first I thought it was just my Facebook app going crazy, but It didn't take me long to realize it's everything. I've chalked it up to being the update.
I really hope they fix it soon; I came to the platform to get away from things like that and would like to enjoy a seamless experience.
My regular 6 has been having issues since I updated the software to 9.0, and then the two following updates as well. It freezes, apps crash, it's slow, you name it. Done a hard reset countless times and wiped it and started over as a new phone. Works fine for a brief time then starts the issues again. I've started using my 6 Plus as my daily device and it has none of those issues. My regular 6 was great before the new software.