I'm a bit concerned that perhaps your undergraduate requirements weren't stringent enough... You should look at a variety of "typical undergraduate preparation" pages from the websites of graduate programs that you are interested in. I recommend this mainly based on your note regarding Thermodynamics (typically it's recommended that the applicant have one semester of thermodynamics with statistical mechanics). I also personally tend to think the recommended coursework is a bit short... often you see one term of classical mechanics and one term of quantum mechanics... but it's really best if you have two terms of each. Furthermore, many undergraduate programs only "require" one term of EM... and you'll definitely want TWO. Committee members look at transcripts and bump down an application if upper-level undergraduate coursework looks weak. They want to know that you can make it through graduate coursework, qualifying exams and comprehensive exams.
If all your coursework looks well according to these standards, then I would encourage you to branch out into upper-level coursework in a complementary field (like math or chemistry)... or graduate coursework. Committee members do tend to bump up applications a bit for these reasons, especially if the research area you are interested in is multidisciplinary, as many research areas are.
Completing independent study under a professor is one of the best things you can do. Preferably this would take the form of a directed research project (typically under the research program of a faculty member, perhaps underneath one of his/her graduate students). Graduate selections committees do ESPECIALLY look for research experience, because they want to know you'll be a productive graduate student. Furthermore, a directed project under a faculty member will hopefully mean you'll have at least one VERY strong recommendation letter... most graduate program want three letters, some even four. You'll also have something to talk about in your personal statement, and better justification to yourself and the committee regarding the field of research that you plan to pursue.
Studying a topic on your own, while perhaps edifying, leaves no trail of progress. If you attempt independent-study on a topic, I suggest you find a faculty member to actually sponsor the coursework... you can register for it, get a grade, and get a possible recommendation from the faculty member. Sometimes nearly-emeritus or emeritus professors are the best for this type of activity. They're older and often not as active in their research programs... and they like to have a young, ambitious student come and discuss topics with them.
I'm also a bit concerned about the statement regarding "retaking all the classes" that you "made mediocre grades in." Hopefully the grades aren't that bad... especially the grades in your field. You may want to look at some forum threads that address the topic of grades and graduate admissions regarding grades. I never replaced a grade in a class via course forgiveness, and I'm not sure how it shows up on a transcript because I'm not sure I ever saw this on any of the applicants in the graduate admissions process that I was involved in.
Hopefully all this helps you better view your options for the next year. Best luck!