Q: How do I select "none" for my payment option

Is this a new account, or one that you've payment details on ?

If an existing account, and if you don't iTunes any money and aren't a family organiser in family sharing, then try tapping on your id in Settings > iTunes & App Store and then tap 'view apple id' on the popup and log into the account and see if you get the 'none' option : Change or remove your Apple ID payment information - Apple Support

If it's a new account then you need to use the instructions on this page when creating an account : Create an iTunes Store, App Store, or iBooks Store account without a credit card or other payment method - Apple Support - if you don't then payment details will need to be entered before you will get the 'none' option