A simple and practical apparatus enabling repetition rate (f_rep) noise, carrier-envelope frequency (f_ceo) noise and n^th optical comb mode (ν_n) noise spectra measurements with high precision is established. The f_rep and ν_n noise spectra are measured by a fiber delay line interferometer, while f_ceo noise spectrum is measured by an f-2f interferometer. We utilize this apparatus to characterize the noise performance of an Er-fiber optical frequency comb (OFC) and analyze the origin of dominant noise sources. Moreover, this apparatus provides a powerful tool for diagnosing noise dynamics intrinsic in mode-locked lasers and OFCs. To this end, we uncover the anti-correlation between f_rep and f_ceo noise as well as the impact of servo loops on noise characteristics in the stabilized OFC.