Thank you for jumping in Leandro and Andrew. As it turns out my problem had to do with the fact that my system is not built as a cluster, but is a small, all-in-one system.
Marc54g on Elasticsearch's slack channel got me pointed in the right direction. I needed to add the line discovery.type: single-node to my elasticsearch.yml file, as well as change the network.host entry to 0.0.0.0.
After that the data just started pouring in. I just needed to walk through setting up the index patterns and Viola'! I have data to search!
Thanks again to Leandro and Andrew for stepping in to help, and to everyone who read my request for help as well.