The cathedral is a Gothic masterpiece. Make sure to pay special attention to the main facade, the western porticoes, the west rose window which circles the Virgin in a multitude of colors, and the Gallery of Kings representing the 28 kings of Israel.
Other highlights are the Portal of the Virgin, surrounded as she is by saints and kings, and the flying buttresses which guard the cathedral’s objects of worship. Once inside, make sure to check out the paintings by Le Brun, the rose windows, the statue of Notre-Dame and the statues of The Piety and Louis XIV behind the altar.
Climbing the towers is one of the most popular activities at Notre-Dame. Access to the cathedral is free and there isn't usually a long wait, especially early in the morning. You're allowed to take photos, but flash is prohibited (it's quite dark on the inside, and the photos don't always come out well).