The sentence you gave is mostly correct but can be made clearer and better with a few small changes. Here is a corrected version:
Original: Swans taxi their chicks around until their soft fluff changes into real feathers and the chicks can swim on their own.
Corrected: Swans carry their chicks around until their soft fluff changes into real feathers and the chicks can swim on their own.
Explanation: The word 'taxi' is usually used for airplanes moving on the ground, so 'carry' is a better verb to describe swans moving their chicks. The rest of the sentence is good and easy to understand!