After leaving the shop you'll both encounter the mysterious Lutece siblings. Elizabeth will ask you for making a choice of a pin and you can choose between a cage and a bird. From now on Elizabeth will be wearing the pin you've picked for her.
Explore the remaining stands on the promenade and then proceed towards The Arcade.
The Arcade is closed and therefore you'll have to ask Elizabeth to brake the lock on a door located to the right of the main entrance. This is the first lock encountered in the game and only the girl's hairpin is required to open the door. You should know that only story-related locks are so easy to pick and all the remaining ones will require to have a sufficient number of lockpicks.
Explore the office you've just entered and then enter the corridor leading to the arcade. Before getting to the main area you'll find a Dollar Bill machine. Check the corridor located behind this machine, because there's a small room for employees at the end of it. There's Gear #4 - Legs on the table and Voxophone #18: Heaven - Daisy Fitzroy in the basket.
There may be a surprise waiting you in the last corridor leading to the arcade. If you didn't throw a ball at an imprisoned young couple in chapter 3, then you'll meet them here and they'll have a small gift for you: Gear #5 - Head. Search the nearby "Colored & Irish" washrooms and then enter the arcade.
Make sure to explore the arcade, collecting mostly money and groceries. There are toilets for "white" people opposite to the corridor you've used to get here. The toiled on the left (the one with the sleeping servant) has more valuable treasures, including Voxophone #19: A Dog's Loyalty - Zachary Hale Comstock.
Once you've searched the washrooms go to where the corridors are separating. You'll find Vantage Point #13: Kinetoscope - Solving the Irish Problem here.
When you're done exploring the arcade proceed towards the entrance to the Gondola Station. This is a one-way passageway so make sure that you've performed all side activities in previous areas. Use the corridor to get to the ticket hall, but before entering it explore the lower floor located behind the hot-dog stand.