I agree with Aiden Grossman that perhaps you should try another algorithm before messing with NEAT, as NEAT is fairly complex. However, I thought I might explain the benefits of NEAT as they pertain to the second part of your question.
The NEAT method is from a paper written by Kenneth O. Stanley and Risto Miikkulainen titled Evolving Neural Networks through Augmenting Topologies. NEAT is interesting for a number of reasons, but one of the biggest advantages of it is that the topology is dynamic. What this means is that the number of neurons in each layer and the number of layers changes as we move through generations. We do not even choose an original number of hidden layers and neurons as we want to start with a minimal topology to avoid having portions of our neural network that haven't had to withstand any testing.
Again, I don't believe NEAT is a great starting point for understanding machine learning. Perhaps you might want to look at binary classifiers, decision trees, or the perceptron to gain a general understanding of the field before progressing to more complex methods.