What is Rapunzel looking for throughout the movie?

She is looking to be free and be out of the Tower to see the lights or the stars

Throughout the movie Tangled, Rapunzel is looking for the lanterns that are released in the sky every year on her birthday.

Explanation:

Since Rapunzel (who was a princess) was kidnapped as a child by Mother Gothel, her parents (the king and queen) release thousands of lanterns each year on her birthday, hoping that their child would return home.

For 18 years on her birthday, Rapunzel saw the lanterns floating in the sky from her tower, where Mother Gothel kept her. She always wanted to go see them, but Mother Gothel said they were just "stars" did not let her.

She later forces Flynn Ryder to take her to them after he broke into her tower.