Hi! I recently graduated from UCLA Applied Math. Full disclosure: I also had a stats minor, and technically I was in mathematics of computation. I am now a grad student in Computer Science at USC.
Many of my friends went to grad school. I know that's not a career, but it will hopefully end well for them, and they can be researchers or professors (hopefully both).
Many others are software engineers. If you have significant programming experience, many engineering companies and technical firms may offer you positions to do scientific programming (DE solvers, fluids simulations, structural simulations, Other Things for the defense industry, etc). Several of those software engineers are at non-math related companies.
Outside of that, I know a good number (20-30 out of a class of several hundred) who are analysts. This is a essentially a statistical position, at least for the analyst positions they're employed in. Some of them are in marketing, others in finance, but all do about the same thing; they push around a lot of data.
None of them have the title Data Scientist, but they might get there at some point. And lets be honest, Data Science is stats with added...well, I'm not really sure what they add. But it's very similar.