Sorry this is delayed a few days; I was working hard late last week on a few things.
So, in the past few days we've talked about the traditional ways to enter a career in astronomy, from things to consider about the career to high school and college preparation. Let's say you've finished these. What's next?
The traditional career path is in academics. Once you have a bachelor's degree, you continue on to graduate school to earn a Ph.D. in astronomy/astrophysics. Then you become a "postdoc," a researcher in a temporary position (usually three years; this is the stage I am at), and finally on to become a faculty member. Easy, right! :)
There are many other career paths these days, though, and often you can wander from one path to another. The government hires many astronomers. The most famous is, of course, NASA, which has many astronomy research facilities across the nation. The astronomers there are typically focused on a certain air- or space-borne observatory or research program. The government also hires astronomers at research laboratories, such as Los Alamos; often these astronomers are working on research programs that can have impact on national defense as well as astronomy, such as supernovae, which are the explosions of giant stars and involve lots of understanding of nuclear physics and nuclear explosions. Finally, the government also runs a few national astronomy centers, such as the National Optical Astronomy Observatory and the National Radio Astronomy Observatory. Jobs at these facilities often involve a little research, a little management of telescopes and facilities, and work on the design and construction of new telescopes.
There are many jobs for people with training in astronomy in private industry. I am not very familiar with these aspects, but I know that jobs range from work on satellites (often with optics systems and guidance systems) to production of research-grade telescopes. So, there are many jobs for the professional astronomer!