What are key things to test in ArcMap 10.1, service pack 1?

I would suggest testing those things that most of your users may use. The service pack Issues Addressed page is grouped by application theme (e.g. ArcGIS for Desktop, ArcGIS for Server...etc) to help you test against.

The big new features at 10.1 SP1:

• 64-bit background geoprocessing
• Parallel processing factor environment setting on certain tools
• arcpy.da.Walk function
• Better JSON, WKB and WKT support in arcpy
• Some improvements to the (still) rather annoying ArcGIS Connection Utility
• Various improvements for ArcGIS for Server

A more easily digestible list is here: What's new in ArcGIS 10.1 service packs

Specifically for geoprocessing: What's new in geoprocessing in ArcGIS 10.1 service packs

One major update with ArcGIS 10 SP1 is 64-Bit Background Geoprocessing, available as a separate download here. There are major implications for any organization that process big-data:

This 64-bit installation (which requires a 64-bit operating system) will replace your current background processing. All scripts, models and tools which execute in the background will use a 64-bit process. Your biggest benefit when using 64-bit Background processing is that using 64-bit applications allow your large processes to take advantage of more memory (if available) than when using a 32-bit application.

I think you will find a lot of interest in this new feature and it would be easy to incorporate this into a presentation. I've personally had great success using 64-bit background geoprocessing on memory expansive operations such as Integrate, where 32-bit processing had previously failed.