A general introduction to neural networks (it seems you still need to learn a bit what they are): http://www.doc.ic.ac.uk/~nd/surprise_96/journal/vol4/cs11/report.html
Read this document which explain how feedforward networks with backpropagation work (maths are important): http://galaxy.agh.edu.pl/~vlsi/AI/backp_t_en/backprop.html
Here you have one implementation in matlab with comments: http://anoopacademia.wordpress.com/2013/09/29/back-propagation-algorithm-using-matlab/
Regarding your questions:
1) "When creating a neural network, do you simply define the amount of nodes and have each node in layer i connect to every single node in layer i+1?" Depend of the network you use. In simple fully-connected feedforward neural nets, yes.
2)"And then simply have them learn the weights themselves?" That's the general idea. You have some data of which you know their classes (supervised learning), which you will give to the neural network to learn the pattern, and after learning is finished you use this updated weights to classify new, unseen data.