The intent of this question is to provide a list of learning resources for people who are new to generating functions and would like to learn about them.
I'm personally interested in combinatorics, and I sometimes use generating functions in answers to combinatorial questions on stackexchange, but I know many readers are not familiar with these objects. I hope this list will help those readers and provoke interest in GFs generally.
I will provide an answer below, but feel free to edit my answer or provide your own answer. Initially it will be a short list, but maybe it will grow over time. Please regard this question and its answers as a community resource.
Acknowledgement: In asking this self-answered question I was prompted by helpful advice provided by this discussion on mathematics meta stackexchange, users quid and John Omielan in particular. Thank you.