Doug,
So there are many ways for you to learn about ArcGIS.
Overview of help for ArcMap
ArcMap | ArcGIS for Desktop
and for geoprocessing and automation, a good place to start.... (notice the python and arcpy topics too)
What is geoprocessing?—Help | ArcGIS for Desktop
Since you said you free trial ran out, maybe you would be willing to spend $100 for a year for a personal (non-work, non-profit) that you can install and play with at home....full "Advanced" level with many of the most popular extensions.
ArcGIS for Personal Use Program | ArcGIS for Desktop Advanced for Personal Use
On the developer's side of things...
ArcGIS for Developers | ArcGIS for Developers
The Esri Developer Network (EDN) annual subscription is how you would access the server side of things, but a free "developer" account gives you access to the ArcGIS Online stuff for testing/developing.
For training, lots of free or cheap resources...
Esri Video | Videos Covering GIS Events, Products, People, & Topics
Esri Training
arcpy specific resources from one of the classes above Introduction to Writing Geoprocessing Scripts with Python 10.2/10.3 Class Resources
So there is a list of things to check out. But as Chris asked, if you want specific info, you may need to give us some idea of what you are trying to automate. That will help us give a more focused answer. But most of all, have fun with it....GIS is fun.