By the way if you want to explore network science but don't really know where to start, consider Gephi (currently being refactored, and just updated a few days ago) or Cytoscape (if you're more drawn to bioinformatics).
Both make it easy to load/generate standard datasets, import tabular data, and have a good selection of plugins. It's easy to kick stuff out to either from NetworkX using /gefx or graphml, and you'll be able to experiment with a wide variety of layout algorithms and metrics. If you don't find the toy/benchmark networks intuitive, considering hitting the HN API for your source material; it's a lot easier to grasp the topic by studying relationships that are already familiar.
Do you have any advice/ideas about ways to learn about graph layout algorithms themselves, including dynamic/real-time algorithms (which allow for user interaction)? I have been skimming through various papers and the first book on this page [0], in particular the chapter on force directed algorithms, because they seem to be the earliest and most general graph drawing methods.
You mean algos related only to NetworkX or in general? If you are looking for NetworkX related stuff besides the official docs, NetworkX Guide [1] is a good starting point.
In general! Graph drawing and interactivity is an interesting problem in my opinion.