...analyse musical notes and find the fundamental frequency to determine the pitch.
So, measure 440 Hz and output "A" (?), a tuner application.
To do this I'm reading in audio data, taking an FFT, taking the auto-correlation of that FFT and then finding the IFFT of that.
Having done that I then find the maximum peak (which seems to always be a zero) and then find the next highest peak and determine the lag between them. Finally I calculate the period from this and find the fundamental frequency.
This sounds like the superposition of two methods and two domains.
In the above process, once you obtain the FFT, you "transit" from the time domain to the frequency domain. In this domain, your signal is expressed as a sum of sinusoids at different frequencies.
Therefore, all you have to do (in this frequency domain) is locate the highest peak sinusoid.
This might be very easy if you are recording an electric instrument (bass, guitar, other) over wire or very challenging, if you are recording an instrument over the standard microphone found on a mobile phone.
BUT!, instead of using the FFT first, you could obtain the autocorrelation of the signal and then examine if that autocorrelation is periodic. The added benefit of this is robustness to noise.
Assuming that a dominant sinusoid exists in a recording, measuring the distance between the first and second peaks of the recording's autocorrelation would give an estimate of the period of the sinusoid. This technique applies to the time domain.
Because of the noise aspect, you might have to use a combination of the two. So, for example, record a signal, take its autocorrelation then apply FFT to the autocorrelation to discover its periodicity. In parallel to this, you might want to estimate spectral flatness as a measure of quality because it might be that the signal at the input does not contain any periodic component.
Finally, please note that due to the way discrete autocorrelation works, the autocorrelation of some signal $x(n)$ composed of $N$ samples, will have a length of $2 \times N - 1$ samples...which would need to be taken into account when trying to derive timings and frequencies from it.
Now, this is as far as what is mentioned in the question is concerned. In general, there are various techniques to perform pitch detection ranging from the very simple to rather complex ones. For a brief overview, please see this link, this link and this link, while, for something more specific, please see this link and this link
Hope this helps.