Throughout the museum are various plant boxes you can water using your farmer outfit. Water all of them for the shield piece.
In the first area (before entering the dinosaur exhibit), look for five t-rex posters. Destroy all of them.
In the dinosaur exhibit, use Rex Fury (during Free Play) to pull two handles near the stairs.
In the dinosaur exhibit, go to the second level and solve all four exhibit puzzles. When you solve the fourth one, you’ll unlock the shield piece.