Infinity Blade I
To enter the mode in Infinity Blade I, you must obtain The Infinity Blade. Use it to unlock the obelisk in the dungeon, and kill the three prisoners to unlock the large door. Proceed through it and kill both Zero Mech and Archarin. An option will appear to enter NewGame+.
When you enter NewGame+, you will lose all your items and gold. However, you will be able to master all the items again, to a second level. This feature persists even if you haven't mastered an item from the Original Game mode; i.e. you'll be able to master the item twice.
If you buy The Infinity Blade, defeat the three Deathless Kings, Zero Mech and Archarin, you will be able to enter NewGame++. Here you can master your items to a third level. You can repeat this process to enter additional game modes and unlock additional Mastery levels. You will not have to repeat the tutorial on blocking during any of these modes.
Note: If at any point during NewGame+ or its derivatives you click on Reset to Bloodline 1, you automatically start at the beginning; i.e. the OG mode.
Infinity Blade II
To enter the mode in Infinity Blade II, you must kill Raidriar. Afterwards, you can go to "Character Slots" in the settings menu and enable NewGame+. The NewGame+ save is separate from the original game save. The game mode can be switched at any time by changing character slots.