Back up to iCloud failure

My new iPhone 6s just arrived so I excitedly tried to set it up using a combination of strategies, including suggestions from apple support, this community and my service provider, all of whom offered conflicting instructions. And none of their suggestions worked.

Apple support told me to do the following:

Connect my existing phone to power, lock the phone, turn on iCloud Photos and enable iCloud Drive, then turn on auto backup in settings and manually back up. Several hours later I checked the phone and the backup had failed. Tried again, same result.

My existing phone is a 5s running iOS 9.0.2. It has never been backed up to iCloud before, only to my Win8 laptop. Never had any issues doing that. And it only takes a minute or two.

Prior to this, I had iCloud Photos OFF. My photos taken on my phone showed up on my iPad without iCloud photos. All that was enabled was Photostream. Photos synced across my two devices without me doing a thing. When I called apple support I was told that this is impossible. Well it was working for me! I don't have a lot of photos and storage was never an issue. Every photo I have on my phone is already on my iPad. So what I want is to basically use restore to copy everything on my 5s onto my 6s. However the backing up part has proven impossible.

TMobile told told me I had to swap SIMs to move my data and settings and that a restore from iTunes would not work since my contacts, calendar events, notes and reminders do not get backed up to iTunes only to iCloud. I have already used iCloud for those apps from the days of iOS 5.1. All that data is on my iPad.

So how do I use restore to set up new phone if I cannot get it to back up to iCloud?