Weird Situation: Pre-med to Quant Switch

I'm currently a rising senior at a non-target(semi-target?) school. All my life I've been working towards the promised land of medical school, but about a month ago, I had the sudden revelation that maybe I'm not really into this anymore. After finding out about quant finance and algo trading, I just felt like these suited my interests and personality a lot more than medicine did.

That being said, I'm obviously a bit late in the game. However, I am working toward a degree in Math, along with a Neuro degree, and currently have a 3.9 GPA with non-financial EC's. I also know Python, R, and SQL well, although I don't have professional experience with any of these. I plan on taking some classes in Financial Derivatives, Stochastic Calculus, and (hopefully) Machine Learning this coming year. Yeah, I'm a nerd whatever.

Getting to the question: do you think I have a shot at a quant internship next summer in an IB/hedge fund? If not, what other internships (data science?, etc.) should I pursue to maybe break into the quant field in the future? I'm just kinda lost right now; what's the first step I should take to developing algorithms and trading at an HF?

Also, do I really need an MFE or PhD to get into quant? Because my research says FUCK YES. I feel like most of the stuff in an MFE are not too hard to learn with a few books, so if I hustle like hell and network enough I should be good?