After a little (well a ton really) more testing, I really think it has something to do with a setting that is being carried over from a previous iOS version when you upgrade to 9. I upgraded to a 6s yesterday and restored the latest backup from my 6 to the new phone. At that point, I had not deleted the old data and settings from my 6, and both phones had the problem. I could delete my gmail and icloud account a hundred times and re-add them, but the calendars just would not sync. My gmail email would, but no calendar or contacts. So, I deleted everything off of my 6 and started from scratch. At that point, sync worked perfectly for calendars and contacts. Keep in mind that both are on iOS 9.0.2, so it is not inherently something wrong with the OS itself, but something that is carrying forward from settings or apps in a previous iOS version when you upgrade to iOS9 with data and settings already on your phone. If you start all over with just the iOS9 on the phone, everything is okay.
So I did what was the last thing I wanted to have to do.... I started all over on my 6s. Took it back to factory and added my data and apps back to it manually rather than through a restore process, which would have just added whatever corrupt setting thats causing all this to begin with. Now, about 8 hours later, all 188 apps and all my data is back and working fine. The only thing I lost (thanks to all the cloud syncing options in most apps) is my old text messages. But I was able to use a 3rd party program to back them up. My 30GB of photos luckily were in Google Photos as well, so I can move back across into my photo stream any I want local on the phone rather than in the cloud.
Hopefully Apple will figure this out, but I bet not. There just doesn't seem to be enough people with whatever setting or app on their phone that is carrying forward into iOS9 and causing this to happen. Maybe I am wrong, but good luck.