Orbital eccentricity is one of the basic planetary properties, whose distribution may shed light on the history of planet formation and evolution. Here, in a series of works on Planetary Orbit Eccentricity Trends (dubbed POET), we study the distribution of planetary eccentricities and their dependence on stellar/planetary properties. In this paper, the first work of the POET series, we investigate whether and how the eccentricities of small planets depend on stellar metallicities (e.g., [Fe/H]). Previous studies on giant planets have found a significant correlation between planetary eccentricities and their host metallicities. Nevertheless, whether such a correlation exists for small planets (e.g., super-Earths and sub-Neptunes) remains unclear. Here, benefiting from the large and homogeneous LAMOST-Gaia- Kepler sample, we characterize the eccentricity distributions of 244 (286) small planets in single (multiple) transiting systems with the TDR method. We confirm the eccentricity-metallicity trend whereby the eccentricities of single small planets increase with stellar metallicities. Interestingly, a similar trend between eccentricity and metallicity is also found in the radial velocity sample. We also found that the mutual inclination of multiple transiting systems increases with metallicity, which predicts a moderate eccentricity-metallicity rising trend. Our results of the correlation between eccentricity (inclination) and metallicity for small planets support the core accretion model for planet formation, and they could be footprints of self (and/or external) excitation processes during the history of planet formation and evolution.