I want to prove to grad schools that I can think and learn by myself.
Students take CLEP exams for many reasons. From the above statement, it sounds as though you expect it to strengthen your application. It won't. The kind of independent work graduate schools look for is research -- an honors thesis, an independent study in your major, working in a lab. They want to see advanced work that goes beyond the classroom. If you have to choose between studying for a CLEP or doing independent research, always choose the latter.
But will the credits count? Graduate schools want to see most of your relevant credits (that is, in your field) taken at a four-year university. Because uniformity of opinion doesn't exist across all programs, some may find your CLEP credits intriguing -- as long as you don't rely on them -- and others may dismiss them. You do want to be careful about using them as required courses for graduate school without following up with a more advanced course. For example, if a program's admissions requirements requires a year of calculus, and you take the CLEP without taking calculus II at your college, the adcoms may think that your preparation is shaky.
IIRC, most of the CLEP exams cover introductory courses/topics, so seeing them on your transcript will be more of an curiosity than help or hindrance.