This achievement is awesome.
First, you start with a fish as soon as you create a character. Feed it to your pet.
If that's not to your liking, there are several pools that you can go up to and hit A to fish. when the circle turns red hit A again and you'll catch a fish.
Simply go to your player screen. Select the fish and hit A. It will atomatically feed it to your pet. Transforming it to somethign awesome and netting you an achievement. :)
side note: If a fish says the transformation is "permanent" don't let that fool you. If you don't like what your pet becomes, feed it a different fish and when the time runs out it will change back to its original state.