The sentence you wrote is almost perfect! Here it is with a small improvement for clarity and correctness:
"Lyrebirds are great singers and can mimic noises made by birds, animals, and even machines."
The only change is adding a comma before "and" in the list (between "animals" and "even machines"). This comma is called the Oxford comma and helps make the sentence clearer.
So, your sentence is grammatically correct and easy to understand. Great job!