Objective 1: Escort the Little SisterOnly a Little Sister can unlock the doors in the Proving Grounds so that you can reach Fontaine. Keep her alive as long as you can. If she dies, summon another Little Sister from any glowing vent by banging on it with your Wrench. 
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The Little Sister is very excited to head through the museum that serves as the Proving Grounds for newly inducted Big Daddies. The Proving Grounds consist of a circuitous path through a museum crawling with Splicers, Turrets, and Security Cameras. There are several stations where the Little Sister must stop and extract the Adam from a target corpse and it is during these times that you must really work hard to protect her. Keep the Incinerate 3 plasmid ready in your left hand and the Machine Gun in your right. The Antipersonnel Auto Rounds should serve you well here.
Follow the Little Sister into the Museum Lobby, but let her run up the stairs ahead of you for the moment. Head down the path to the left to quickly explore the Gift Shop—the Circus of Values machine there carries most types of ammunition.
Follow the Little Sister around the balcony of the Central Atrium (you can quickly explore the small side-room to the north) and through the door to the left. Continue down the stairs into the South Wing.