My physics background is sparse--I only know classical mechanics and E&M, and I have a self-taught (and therefore foggy) quantitative understanding of special relativity to go with my (I think) strong qualitative understanding. My math background is a bit better but not much: multivariable calculus, linear algebra, differential equations, and after this Spring I'll have some abstract algebra under my belt too. I want to figure out if MTW would be a good text for me, but I can't find anything other than its table of contents online, so I have no idea how much it assumes understanding of differential geometry and how much the book teaches you it. If MTW is too advanced for someone of my math-and-physics background, is there another text you would recommend? And I'm perfectly okay with the answer "Please just wait until later before trying to teach yourself GR." If that's true, that would be the most helpful response.
MTW is too advanced. You could try Hartle; all you really need is Lagrangian dynamics/calculus of variations and some orbital mechanics. (If you don't know these just read a few chapters of an intermediate mechanics book--Taylor is my favorite--and you should be OK.)
Your background seems fine to me for reading MTW. However, MTW is expensive and 40 years out of date. I would suggest Carroll. It's much more recent, and there's a free version available here: http://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll_contents.html If you're concerned about your SR background, try Spacetime Physics by Taylor and Wheeler.