Most intro classes just require familiarity with basic calculus (differentiation, chain rule), linear algebra and basic probability all of which you can just lookup directly on https://www.expii.com for a short tutorial. Toolkits are usually in Python or Lua, plus the numerous textbooks 'Deep learning with python' that are around and specific DL books such as http://www.deeplearningbook.org/.

Afterwards look around for Adversarial Learning, like detecting perturbations that force mis-classification and other attacks described in papers by Carlini and Wagner. Currently there isn't a perfect defense developed for all of these attacks, except robust optimization that provably defend some of them. Attacks are an interesting area in DL you can get into since we don't have access to large resources and can only do DL on a small scale (in my case anyway).