How to move a Reminder from one list to another in iOS?
Using iOS 6.1.2 it is pretty straight forward (I have no older version of iOS with me to check how and if it works there, but I guess it will).

Tab on the task/reminder you would like to move to another list. Choose more options or show more (my language setting is German so I'm not 100% sure how this menu item is called in english). There appears an option titled List. This will allow you to move the item to another list.