The primary way in which administrators expand their base is by building onto it; that is, constructing additional rooms and hallways. To do this, simply use the "Room" tool under the "Construction" menu and click-drag out a new room that connects to the rest of the complex.
To expand a preexisting room, use the Room tool to draw out the expanded dimensions of the room, and then have your Builders tear down (not vaporize!) the dividing wall between the preexisting room and its expanded section. The unzoned section will automatically copy the room's zone.
Note: You should not mark the dividing wall for demolition until the new section of room is built and oxygenated. Builders will sometimes destroy the wall between the two sections before the new one is sealed, exposing the room to vacuum.
The following are useful tips for expansion.
As a base grows, so do its Power, Oxygen, and maintenance demands. Eventually, you may have too large of a base for your Technicians to keep up with maintenance, which is a very bad situation if Raiders attack. Just like a real-life space program, you must ration your rooms, resources, personnel, and defenses to survive. Make every square of floor and section of hallway serve a purpose, even if that purpose is just to eliminate the claustrophobia of your citizens. Do not build more Reactors than you need, nor more RevivaBeds or Weight Benches or any other object. Overexpansion is an easy way to lose the game.
It is possible to build outposts that citizens will use, but these outposts must have their own power supply, oxygen, airlock, and other necessities. Building outposts is an advanced strategy that is not recommended for beginner administrators. These outposts can become permanently staffed by citizens by assigning them a bed inside that outpost. Be aware of the following hazards when building an outpost: First, a fire turns an ill-prepared outpost into a death trap. Numerous crew will arrive to fight it and deplete the outpost's oxygen in the process. Most suffocate in the resulting panic, with others being sucked into space through the airlock. Second, a strong Raider attack can obliterate an outpost, depriving you of precious crew and rooms that could have been used to defend the main base.