In order to get started with neural networks, begin with drawing simple neural nets for basic operations like addition, multiplication, XOR. Just represent boolean tables as neural networks.
Once you can do that, move on to implementing the algorithm yourself. A simple 3 layer network is enough to understand how the concept works. 4/2/2 nodes is plenty. Just understand how the calculations work.
Then move on to a framework - only after you understood the math. The machine learning course on coursera by Andrew Ng(?) explains the algorithms.
A single layer isn't a "neural network". The difference between logistic regression and a really simple neural network is the ability to behave in a non-linear way.
A single layer amounts to a matrix multiplication. Matrices are linear operators.
2. mnist hand written digit recognition
Here is the code..
https://bitbucket.org/sras/haskell-stuff/src/b58f3fc017ce303...