My phone will not let me delete a payment method and add another one. I just keep greeting some kind of red error message.

Sign in to your Apple ID account page and edit / update your Apple ID details -> https://appleid.apple.com                                                     

• On the Sidebar click Payment methods -> then tap Payments Methods and tap Manage payment method.

Or see -> Add a payment method to your Apple ID

Add a payment method to your Apple ID – Apple Support (UK)

• If you want to change your card, you need to add first the new card, then remove the old card.

See also the following Apple articles:

Payment methods you can use with your Apple ID

Payment methods you can use with your Apple ID – Apple Support (UK)

fyi: Not all credit & debit cards seem to work with App Store / iTunes Store, try a different one, or preferably a credit card. If your card is not accepted as a paymenf method, you need to contact your bank.

Remove a payment method from your Apple ID

Remove a payment method from your Apple ID - Apple Support

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If you can't remove a payment method

If you have subscriptions like iCloud+, have an unpaid balance, or you use purchase sharing, you must keep at least one payment method on file.

• Do you have a subscription? You can cancel iCloud+ and cancel other subscriptions from Apple. When your subscription period is over, remove your payment method.
• If you're in a Family Sharing group and you use purchase sharing, the family organizer must have a payment method on file. If you're the family organizer, turn off purchase sharing, then remove the payment method.
• If you have an unpaid balance, you might not be able to remove a payment method. Instead, change your payment method.
• If you still can't remove a payment method, contact Apple Support.

If you need to change or update your Apple ID payment method

If you need to change or update your Apple ID payment method - Apple Support

If you still need help, get more help and assistant from Apple directly, see Contact Apple for support and service -> Contact Apple for support and service – Apple Support (UK) 

Make sure you are adding the other payment first before you delete the current one.