Here is a shot of the Nano inserted into the IO shield. This was after troubleshooting, and as you can see, the LED's were good to go.
In the web IDE at https://www.arduino.cc there is an example called ReadASCIIString in the Communication section in the examples. It was geared toward a common-anode RGB LED, so I modified it slightly to work with 3 conventional LED's connected to a common cathode, and when I ran it, the blue and green LED's ran with no trouble, but the red one was off. I suspected my soldering skills at first, so I unplugged from the shield, and dropped the Nano into the breadboard. Everything worked. So my thinking was that the shield was bad. Sometimes, it can be harder to spot the most obvious problems. As it turned out, the jumper wire was the issue. I used my Extech multimeter on its Continuity setting to determine this. In case you have one, it's the one marked "CAP" and has an Omega symbol, along with a diode symbol and what looks to be a sound icon. I put a jumper wire into the sixth female pin on the right (where the D3 pin from the Nano should go), and I touched the red wire from the meter to it. I touched the black wire to the "S" (signal) pin marked D3, and the meter read nonzero numbers. So that proved to me there was a problem with the jumper wire. Plugging everything back in verified that was indeed the case.