1) Palmtree Panic Act 1
In the final stretch of land before you hit the goal post, you can find Amy waiting for you. If you stand still and let her come near you, she'll hug you and you'll get your achievement. This must be done in the Present, as Amy is not there in the Past, Bad Future, or Good Future. Also, make sure you don't have a shield on. If you do, Amy won't be able to hug you.
2) Stardust Speedway Act 3
This is much later in the game and is not missable. Right after you win the race against Metal Sonic, instead of the capsule you normally hit to free the animals and end the stage, you find Amy tied up. Jump up and hit Amy to untie her. She'll drop down, hug you, and the stage will end.
Your achievement will unlock the moment she hugs you. Good luck!
