See this link at the American Astronomical Society: A New Universe to Explore: Careers in Astronomy Brochure | American Astronomical Society
You would generally need a doctoral degree for research positions.
With only an undergrad degree, your most common job opportunities would be in high school teaching (physics and/or math). You might find some positions in planetariums, science museums, or science journalism, if you have the right background, though these are fewer in number. Math and computer science coursework beyond the base requirements for an astronomy/physics major will strengthen your qualifications and might open up additional options.
Of course, you could always go into other graduate/professional fields, too (law, science & technology policy, etc).