Want to dive into Python scripting in ArcGIS 10.1 but don't feel like reading a long, involved blog post about how to get started? That's good, because I don't have time to write one!
The Python Window is an easy way to start experimenting with Python in ArcMap. You can click on tools in toolboxes and "drag and drop" them into the Python Window.
The ArcGIS Python Community page has quick links to presentations, videos, tutorials, and other helpful resources. This is invaluable to beginners and experts alike.
If you want to automate repetitive tasks involving map documents (MXDs), for example checking all the MXDs in a folder for broken data layers, you will be interested in the arcpy.mapping sample scripts.
If your work involves more ArcGIS Server administration, then the Server REST API administration scripts will be more useful to you.
Finally, one of the main new Python features at 10.1 is the Data Access module (called arcpy.da). This provides new improved cursor objects to access geodatabases that are more powerful (for example, they can control edit sessions) and much faster, often more than 5 times quicker than the old cursors. The old cursors are still available to support legacy applications.