Computing the autocorrelation function is a robust approach to determine periodicity: the degree of periodicity will be indicated by peaks at offsets $\tau$ in the autocorrelation function. The FFT alone can be used to indicate periodicity of individual tones, but the autocorrelation function can reveal with better fidelity the periodicity of any function including noise like functions or functions that may be buried in noise.
I show an example of this where I synthesized a received GPS signal down-converted to baseband together with the dominant receiver noise. To signal appears as additive white Gaussian noise (AWGN) in both the time and frequency band given the low level GPS signal received. The autocorrelation function reveals the 1 ms periodicity of the GPS C/A code used while we can't distinguish this in the frequency domain. If the captured signal was truly AWGN only, we would only get the single central peak.
A zoom in on the frequency spectrum for this signal (as determined from an FFT) is given below:
The 1 ms periodicity would appear as harmonics spaced at 1 KHz in the frequency spectrum if visible. We see in this case that it is very difficult to distinguish the 1 KHz harmonics from the noise, yet in comparison we were able to see evidence of the 1 ms periodicity much more clearly from the autocorrelation function.
If we repeat the above plots with the GPS signal only and no noise, we get the following result where the periodicity is visible with both approaches:
Each peak in the FFT is a complex number. The location of the peak magnitude on the frequency axis indicates the frequency that was detected with a relative magnitude between various peaks indicating the strength of each component. Often the components will be integer harmonics of what can be described as one periodic waveform—- any periodic waveform that is not s pure sinusoid or single exponential will have integer harmonics consistent with the Fourier Series. The phase of each peak will reveal the relative phase of each component, but this will not be accurate if the frequency component is not perfectly aligned with the center of an FFT bin.