From my own experience researching various mfe programs (and reading these forums for quite some time), the biggest hurdle for non-math majors is a lack of math courses. This includes calculus (at least up to multivariate calc) , ordinary and partial differential equations, probability theory, linear algebra, and usually some combination of stats/numerical analysis. There could be one I'm forgetting as well.
As far as books go, you should do fine with the complete guide to capital markets. Although its quantitatively oriented, not much math knowledge is needed to follow. Also, I feel like I should recommend "how I became a quant" by Lindsay and schnatter, it's a good introductory read that follows 25 quant-type people.
As far as programming goes, I'm really only experienced in c++ and R, so I can't tell you where to go next. It definitely helps though to have concrete evidence of your knowledge by acing a few c++ classes or getting a certificate like the one offered on QN.
Good luck! And if it seems like you have to take a lot of maths, well, you do, but I did almost an entire math degree in just over 2 years so it's by no means impossible. Feel free to PM me if you have any questions.