You can still do exchange programs as a graduate student. That's what I did. I already had a bachelor's degree when I went.

One advantage, if you have citizenship, you can apply for a scholarship to help support your studies abroad (undergrad only). Word on the street is that scholarship is easier to get for STEM majors.

Were I in your shoes, I would be very concerned about going out of the US for extended periods of time while applying for citizenship. I'm not an immigration lawyer, but from what I've heard, that could be a problem.