The other solutions here are great and already contain the list of what you need to do, so I won't repeat that information, but I wanted to offer one tip that can make it less of a headache if you just started playing:
When the game shows you a tool tip such as "Press Y now to dash through the rings!", always do it. If you always follow these tool tips, you will unlock the achievement naturally without any hassle of wondering which ones you didn't do.