Spectroscopically, tidal disruption events (TDEs) are characterized by broad (∼10⁴ km s⁻¹) emission lines and show a large diversity as well as different line profiles. After carefully and consistently performing a series of data reduction tasks including host galaxy light subtraction, we present here the first detailed, spectroscopic population study of 16 optical and UV TDEs. We study a number of emission lines prominent among TDEs including Hydrogen, Helium, and Bowen lines and we quantify their evolution with time in terms of line luminosities, velocity widths, and velocity offsets. We report a time lag between the peaks of the optical light curves and the peak luminosity of Hα spanning between ∼7 and 45 days. If interpreted as light echoes, these lags correspond to distances of ∼2 − 12 × 10¹⁶ cm, which are one to two orders of magnitudes larger than the estimated blackbody radii (R_BB) of the same TDEs and we discuss the possible origin of this surprisingly large discrepancy. We also report time lags for the peak luminosity of the He I 5876 Å line, which are smaller than the ones of Hα for H TDEs and similar or larger for N III Bowen TDEs. We report that N III Bowen TDEs have lower Hα velocity widths compared to the rest of the TDEs in our sample and we also find that a strong X-ray to optical ratio might imply weakening of the line widths. Furthermore, we study the evolution of line luminosities and ratios with respect to their radii (R_BB) and temperatures (T_BB). We find a linear relationship between Hα luminosity and the R_BB (L_line ∝ R_BB) and potentially an inverse power-law relation with T_BB (L_line ∝ T_BB^−β), leading to weaker Hα emission for T_BB ≥ 25 000 K. The He II/He I ratio becomes large at the same temperatures, possibly pointing to an ionization effect. The He II/Hα ratio becomes larger as the photospheric radius recedes, implying a stratified photosphere where Helium lies deeper than Hydrogen. We suggest that the large diversity of the spectroscopic features seen in TDEs along with their X-ray properties can potentially be attributed to viewing angle effects.