I work in the Guidance, Navigation, and Control department for a big aerospace company and also did my undergrad in math/physics. Here's my advice:
(1) Get a Master's in aero, since your math background alone has not taught you how to think like an engineer. A mistake most mathematicians/scientists who become engineers often make is assuming engineering is best treated from the perfectionist mindset of math/science and not the messy combination of technical knowledge, politics, and multidisciplinary cooperation it really is. The master's degree won't get you all the way there, but it will get you used to talking with engineers and thinking like one.
(2) You better know control theory, and well. You need to be really conversant with classical control theory, since this is what most design actually uses. Learn a lot about state space too (I think it will be easier if you have a good background in functional analysis). You can learn about optimal control, adaptive control, robust control, nonlinear control, etc. but keep in mind this is not applied as frequently. Finally, Kalman Filtering is a must since this is the current workhorse for state estimation. As a math student, you'll enjoy the use of Hilbert spaces and stochastic processes used in Kalman Filtering in a way which most engineers won't.
(3) Don't knock implementation. I'm not sure this is what you meant by "if I should just learn how Kalman filters are implemented and how to program embedded C code." but it sort of sounded like this. You will learn that engineers don't really care about math (especially rigorous math) on its own. You need to be able to do something with it. Implementing a Kalman filter on a tiny computer which can fit on a spacecraft and doing so reliably in the often high-radiation environment of space is a field unto itself. There's a lot of math in algorithm development, and you'll learn a ton of physics too!
(4) Don't pigeon-hole yourself. NASA, SpaceX, et al. may be sexy sounding to you now, but that's only because they get tons of press. There's lots of smaller firms, government labs, etc. On a personal note, I wish I had been more aware of how cool defense applications are and studied weapons more. There's a lot of jobs for good weaponeers.
To get started you can get pretty much any book on topics like orbital mechanics/astrodynamics (Bate, Mueller, White [BMW] is the classic, but there are many others), attitude determination and control (I recommend Wertz for an old overview, Sidi for a relatively easy book with a lot of practical ideas, and Markley & Crassidis for the most up-to-date review on attitude determination algorithms.) You'll also need something on control theory. I took my first course using Murray and Astrom, which is free here. Blakelock's Automatic control of Aircraft and Missiles is a classic overview of aircraft autopilots and its modern descendant is Stevens and Lewis' Aircraft Simulation and Control. If you want a good, classic reference for spacecraft dynamics, I have always loved Thomson's Introduction to Space Dynamics.
Finally, lots of people are posting about GPS here. If you are working on guided missiles, aircraft, or satellites in LEO, GPS can be useful to know about (again, a field unto itself which many general GNC people don't know a whole lot about), but for the missions you're probably thinking of, which go beyond the orbits of the GPS satellites, GPS can't be used. Not to mention the whole field of GPS-denied navigation, which is blooming right now. For these applications, you'll need more general ideas of navigation.