There are a million e.g. number parsing (image to digit) neural network type programs on GitHub. Go pick one in your preferred language and break it apart, and rebuild it, looking up the concepts behind parts you don't understand. After you finish up with the above, look up 'the xor problem' to see a common practical problem (which creating a network to replicate xor illustrates, rather than is) and you'll be well on your way to a nice fundamental understanding, built from the ground up.

One of the most interesting things about this topic is that the fundamental concepts and implementations are all really simple. It's the fact that it actually works that's mind boggling. In any case, the above is not a months like affair - but like one week of dedicated work.