As usual, explore and pick up whatever loot you can find, but there isn't anything particularly important to get until a little later. So just go join the crowd at the stage and get the raffle over and done with. Next comes another decision point where you can either keep the ball or throw it at somebody. Throwing it at the announcer gets you an extra gear later in the game and also arms Booker with the melee weapon, Sky Hook. This weapon can be used to execute enemies by holding the melee button when the skull icon appears above the target's head.
Do this 20 times to earn the Industrial Accident achievement / trophy. Follow the objective arrow while fighting your way through the cops. One of them will drop the Broadsider Pistol. It's best to modify the gameplay options so that the enemy's heath is displayed when you shoot them, as this gives you an idea how much damage you're doing especially from a distance.
Pick up the Voxophone tucked between a pair of cages next to the Vigor vending machine (sells Possession Aid for $50), just before you climb onto the police barge that has a turret on it. Use Possession on the gun turrets to make them yours for a certain amount of time, or you can also destroy them if they're proving too troublesome.
Head toward the door at the end of the area where you'll have to kill a Fireman to proceed. You don't really have too many options at this point so just keep shooting him in the head until he drops, then go grab the Devil's Kiss Vigor from him. Continue further down the street without venturing too close to the flames. Kill more cops and collect every loot before entering the Blue Ribbon Restaurant.