Do I need to do anything after upgrading my RAM? I just replaced a 1GB RAM stick in my 2007-era Macbook Pro with a 2GB RAM stick.  It already had Mountain Lion installed on it.

Is there anything I need to do like configuring something in System Preferences or running a command in Terminal in order to make sure all applications and behind-the-scenes stuff take advantage of the increased RAM?

I'm asking with the assumption that some applications might have made a note of the machine's original amount of RAM when the app was installed, and may still be optimizing itself to the old RAM amount.
Nothing other then checking  to see if the OS detected the change. To do that, click the Apple-icon (in the upper left corner), then About This Mac, then you should see your new RAM amount listed there. If not then I'm not sure how your computer is even working.