I would like to know what kind of jobs involve improving processes inside an organization.
On IT, almost (to don't say all) roles are related at some level to methodologies. Ideally, all roles would be 100% aware of what are the available methodologies, but that's another story...
What other topics could be studied to lead to a job using such methodologies / frameworks?
Being interested to methodologies / frameworks will always add value on your career. I don't believe, however, that there's a silver bullet like get used to Agile and you'll rock. So, I believe that the best way to get in touch with methodologies is what you're already doing, i.e., visiting forums like PMSE :)
If you browse through questions, you'll see a lot of very good discussions about n methodologies / frameworks. Do not bypass a specific question just because you're not aware of that topic. Instead, read it carefully and you'll be already studying!
(A question I'd add myself) Who's supposed to bring methodologies to the table?
If you're about to join a company with already well defined methodologies, you'll hit the ground running. On such cases, the methodology will come from Mark's (very well stated!) path;
If you're joining a company that's starting and is still living in a fuzzy environment, you'll be able to help your project suggesting and introducing best practices. Here, you'll have the opportunity to be the one adding value to the team / project company.