Path to career in bioinformatics? I've been completing various courses on bioinformatics through edX and Coursera, and I've made a strong footing in the material. 

I'm not sure what to pursue next in bioinformatics that would bring me closer to research in the field or employment. 

I don't have bioinformatics at my university and thus am unsure of how to progress.

I have a background in mathematics, chemistry, physics and computer science. So, what would you recommend I do next?
The Bioinformatics Stronghold at Rosalind is an excellent resource for building your understanding of many concepts and algorithms you'll see frequently in bioinformatics and genomics.

I'd also say that, if you can make it work, an internship where you can "get your hands dirty" with real problems is a very valuable experience. This option may not be available to everyone due to a variety of factors, but when it is an option it provides a tremendous opportunity to build experience with little risk to the intern or the employer.