The Subway is a way to exchange Goods for Tourists. They call them Commuters, but they function the exact same way. They'll go straight to your open shops and count as two regular shoppers. The goal On The Right Track! gets you started, when you reach Level 20.
The Subway Entrance requires 500 Goods to create 2,125 Coins and 45 Commuters. The first goal will give you one for free, after that you have to buy them for 45,000 Coins. Each is 5x6 and takes 3x8 materials to complete. They can be sold, at a loss. You are allowed to have up to 20 Entrances at a time. A small subway icon will appear above your Entrance each time it's ready to be used. Clicking it will send a Subway Car to the Subway Stop.
The Subway Stop earns 500 Coins and receives the 45 commuters. It is a large 8x8 building and takes 5x10 materials to complete. One comes for free with the first goal. A second is available for free on the Downtown map. You have to expand to where it is. Only one can be built. Removing a Subway Stop will send it to your inventory. A small exclamation point will appear above the Stop when Commuters arrive. Clicking on it will send them to your shops. (In your inventory, the Subway Stop will listed in "Other".)