I would like to learn how to use Python, but I'm not sure what I would use it for. If anyone could direct me to a place to learn the basics of it, I would appreciate it. I am running an ArcView license so I am limited to what I can do with it?
It's a way to replace almost anything you would regularly do manually with an automated script or tool. Geoprocessing, data management, logging, export PDF maps etc are all useful way to leverage python in a GIS environment.
These are computer science classes that use python, and should help you to learn the basics of python and thus arcpy as a python package. The classes are not about arcpy, though, just the basics of python. arcpy is currently supported with 2.7, so perhaps the MIT course would be best, but the 3x courses will help for what is coming from Esri.