What degree(s) should you get in college if you are interested in working at NASA mission control?
Degrees in mathematics, engineering, and the physical sciences are your best bet.
I just wanted to add some more information. If you want to work for mission control in Houston, I'd also recommend trying to get an internship at NASA. You can get an internship in high school or college or both. You might not get an internship working with the Houston mission control team, but just having the experience and connection will get you closer.
There are lots of positions in mission control. All need a technical degree and receive lots of on-the-job training. However, if you know you would like to be a certain type of mission controller, then you could take more classes relating to that position. For example, the guidance and navigation controller would need more orbital mechanics knowledge.
There are also lots of "mission controls" at NASA. Houston deals with all missions with the astronauts once they have left the launch tower. Kennedy Space Center (KSC) has a launch control center (LLC). I worked at KSC. We used the LCC to control the space shuttle whenever it wasn't in space. We needed to run tests and checkouts to make sure it was working. Then on launch day, KSC is in control of the vehicle until it clears the launch tower. They give the go/no-go for the vehicle. Once it clears the tower, then Houston takes over. And there is a Payload Operations and Integration Center in Huntsville that works with the international space station (ISS). Houston mission control also works with ISS.
All the other unmanned missions that NASA has have their own mission controls. I think almost every NASA center has some form of mission control. Jet Propulsion Laboratory has mission controls for most of the Mars landers and satellites. Goddard has mission controls for lots of satellites and some missions that went to the moon.
Ames Research Center is closest to you have they have been working on mission control design:
When there is not a global pandemic, NASA sometimes gives tours of their facilities or has open house days where they open everything up to the public. You can follow NASA Ames either on social media (if you have that), YouTube, podcasts, or signup for their public events announcement newsletter -- https://www.nasa.gov/centers/ames/about/contact_us.html. Then you can ask employees questions and make connections.
I just wanted to add that you might not get the exact position you want right away. There might not be an opening in Houston mission control or maybe you don't get the position on the first try and that's okay. Keep an eye out for jobs you want and keep applying. I would get to know other mission controllers and ask them questions and advice. Everyone is nice and welcoming. We are all together in trying to keep the astronauts safe and explore the solar system.