After our meeting yesterday, I took the circuit home and changed one of the resistors to give us a larger gain. Previously, the circuit would only amplify our ~50mV signal to ~250mV (all of these values are peak to peak). We wanted our signal to have an amplified value less than (but close to) 5V. I was able to achieve a 2.5V signal, which the code seemed to be able to read without any issues. Here is our final circuit schematic:
And here is a picture of our final circuit
When we met today, we tested our circuit once again with the guitar and got the correct notes! It works! We spent the rest of our meeting working on our final instructable. Here is a video of the tuner in action.