Where do I connect the Ping sensor to (Uno or motor shield)?
Put the motor shield on the UNO, then connect the Ping sensor to some unused pin, as defined here:
http://arduino.cc/playground/Code/Ping
What would I use to connect the different pins on the Ping sensor (would this be a 3 pin cable or some other method)?
Wire would work.
Where on the motor shield do I connect the 2 motors to?
The proper places. See those screw terminals along the bottom of the picture? There's text next to them, and I bet you got an instruction manual.
Where on the motor shield do I connect the external power supply?
Same answer.
is there anything else that I need that I might have missed out on or would I be good to go to run the code?
Well, first you need to write the code...