Determination of periodicity in data and finding mean
To determine periodicity, do an autocorrelation. The autocorrelation will have peaks at time offsets consistent with the periodicity.

You mentioned using the Fourier Transform, if convenient you can do a circular autocorrelation using the FT as follows:

$$Corr = ifft(fft(x) fft(x)^*)$$ 

Where $fft(x)^*$ is the complex conjugate (if you signal is complex).