The first thing to do once you upgrade to 10.1 on your new 64 bit computer is to install ArcGIS 10.1 SP1 because it enables 64 bit background processing, thus increasing your memory usage abilities. Whether or not ArcGIS will use it, I cannot say - I'm running 10.0.
The next thing to do would be to learn about using the In-Memory workspace, which essentially skips the hard disk. See this portion of the documentation. If you ever look at your computer's performance during most ArcGIS operations, they're really bound by your disk performance (low memory usage and cpu usage), so taking it out of the equation can be a huge speed boost. Another way to do this would be to install and configure a small RAMDisk, which can provide a little more permanence to your datasets being in RAM (though my disks often get destroyed on shutdown, so have a strategy to keep your data elsewhere).
As for the rest of it, it depends on what you mean by "increase processing capacity" and what your workflow is like. It may be as whuber commented that much of this is not GIS specific. Different tips and tricks will help speed various workflows. I think there is another ArcGIS performance thread somewere on GIS.SE, but I can't find it right now.
Edit:: I've now been running 10.1 for a while. For those who install service pack 1, make sure to install the separate 64 bit background geoprocessing package from the same page afterward, then set your python file associations back to your 32 bit copy (it'll install a separate 64 bit copy) if you use python.