Most universities' mathematics departments have placement exams for lower-division math courses, so, for example, you could "test out" of two years' calculus, most likely. Testing-out of upper-division courses is rarer, so there're not going to be formal procedures in place, but if you simply talk to departmental advisors, they can get the ball rolling to have your self-education appraised. It's just that there probably won't be formal procedures in place, due to the rarity of the event. Math faculty are more-than-happy to see someone who's had the interest and motivation to read on their own.
It's not clear that doing the GRE math subject test (nevermind the other parts) is a good avenue for everyone. I'm not such a fan of that exam as a predictor or appraisal, in any case. One aspect is that it is a very superficial exam, in the sense that it tests test-taking ability almost as much as substantitive knowledge, since it is multiple-choice. Second, it presents a very stylized picture of "undergrad/pre-grad" mathematics... unsurprising insofar as it has to be shoe-horned into a timed, multiple-choice exam.
I think it's also not clear whether you should "try to do research" in order to "make an impression". _Being_interested_ and being curious is one thing, but the further element of presumption involved in too-easily believing that with modest preparation one has unraveled mysteries untouchable by experts... blah-blah-blah... will not make a good impression. That is, honest curiosity and drive are unqualifiedly good things, not presumptious, not silly, all too uncommon, ... and are the features the math faculty would look for.
One last small point: it is very helpful to have in mind the authors of the books or notes you've read, whether they're physical books or on-line notes, if only because the titles of most such things are toooo generic, while the best authors are well-known, and the specific virtues of their books/notes similarly so.
In summary: just talk to the math department undergrad-intake-advisors first, and they will steer you to the right people to talk to in order to have your situation appraised. Some of the people may be a little skeptical, but mostly they will be happy to encounter someone who has the interest who's taken the initiative to study and think about things on their own.