Space Center Houston is the official visitor center for NASA's Johnson Space Center, 40 km southeast of downtown in the Clear Lake area, and it is the most substantial space museum in the United States. The displays include a flown SpaceX Falcon 9 rocket, a Saturn V rocket (the kind that carried astronauts to the moon), an actual flown lunar module, and the Mission Control room used during the Apollo program, restored to its 1969 state.
The NASA tram tour is the highlight — it takes you through the actual JSC campus, including the current Mission Control operations centre, the astronaut training facility, and (on many days) the Neutral Buoyancy Laboratory where astronauts train underwater in a full-scale replica of the ISS. Allow a full day. The centre is 45 minutes from downtown by car; the drive is not scenic, but it is worth it.