It's surprisingly easy to get Arduino talking to Processing via Serial Communication. You can send sensor readings to Processing and make things happen there or send data from Processing to Arduino.
A good start is the Virtual Color Mixer project. Very simple and quick. Just remember after loading the sketch to the Arduino to shut down the software on your computer before running the Processing sketch otherwise the Arduino holds on to the port and won't let go!
Next step is I want to get a couple of sensors on the Arduino to represent the x,y cords. of a mouse. I'm going to start here: Lucky Larry's Arduino + Processing