I've taken Andrew Ng's Machine Learning class, Daphne Koller's Probabilistic Graphical Models class, Dan Jurafsky and Christopher Manning's Natural Language Processing class, and currently Geoff Hinton's Neural Networks class.

I have spent a lot of time on Khan Academy to learn the calculus. In my experience you can get by with a surprisingly small amount of calculus, but it happens to be a small amount from a high level.

For example, backpropagation is just repeated application of the chain rule. Did take a while to get a handle on the derivatives, but it's worth it.