I decided to start here because it was one of my faction's city but I'm in the building and there's no way out! I've looked through the entire building, every floor, and there's no door leading out side. Can someone tell me how to get out?
Just a note that as you walk around a particular facility for the first time, you will start creating a "map" of that facility in the Position -> Travel screen. If at any time you come to a room or see a room with a special marking on it, you can hover your mouse over it to see its description.
For your purposes and as Orion already said, you're looking for the "Entrance" room, marked as seen in the post above.