Political Science Major Classes?

What classes does a political science major take? And what is international relations/studies, does it fall under political science?

You can look that up on college web sites.
These, for example:
http://polisci.berkeley.edu/courses
http://www.middlebury.edu/academics/ps/requirements

The short answer is yes, it can be considered a branch of political science.
However, colleges manage IR instruction in several different ways.
At some colleges, IR is a subfield/concentration/track for majors in the political science (a.k.a. government) department. At other colleges, IR is a separate major. At Georgetown University, IR is taught in the School of Foreign Service, which is separate from the college of arts and sciences (“Georgetown College”).

https://en.wikipedia.org/wiki/International_relations

You might begin with a course in political theory. You would then be expected to gain an understanding of American politics, comparative politics and international relations, as well as a further understanding of political theory. You could subsequently choose a “track” or full major in one of these, or related, subfields.

In general, be aware of overlapping terms. All of “government,” “politics” and “political science,” for example, tend to describe similar courses of study.