Before you get started
First things first I'm assuming you are running the latest iOS available.
You'll want to make sure you back up, either by
Making sure you have iCloud backup turned on (settings/iCloud/Backup) and tapping backup now
Plugging into your computer and letting iTunes back it up for you
Upgrading to iOS 10
On the 13th iOS 10 will start to roll out. The servers are going to be busy so it might take a while to update, be patient.
- Software Update
You should then see that iOS 10 is available. Download and install.
On starting it up you may be asked to enter your iCloud credentials, don't. You can just skip this entry. The problem with doing this is for some reason iOS updates both your App Store and iCloud login to be the same. If like me you have two separate ones it's a pain.
Good luck and let me know how you get on.