1. Create a new backup of your iPad Mini, either via iCloud or iTunes.
2. If you haven't passed the screen on your new iPhone 5S that asks if you want to restore from a backup or set it up as new, choose restore from a backup and choose the one you just created with the iPad Mini.
3. If you've passed that screen already, go to Settings - General - Reset, and then select "Erase All Contents and Settings". Your device will revert to it's native state out of the box. Choose your language, connect it to your Wi-Fi network, if applicable, and then select to restore from a backup. Afterwards, choose the backup you last made of your iPad Mini.