Where to start on neural networks
No, you should go ahead and learn the maths on your own. You will "only" need to learn calculus, statistics, and linear algebra (like the rest of machine learning). The theory of neural networks is pretty primitive at this point -- it more of an art than a science -- so I think you can understand it if you try. Ipso facto, there are a lot of tricks that you need practical experience to learn. There are lot of complicated extensions, but you can worry about them once you get that far.

Once you can understand the Coursera classes on ML and neural networks (Hinton's), I suggest getting some practice. You might like this introduction.