Generalized quartic dispersion Kerr solitons (GQKSs) represent a unique soliton family, which are solutions to the generalized Schrödinger equation with the interplay between the Kerr nonlinearity and the combination of quadratic and quartic dispersions. The well-known pure quartic solitons (PQSs) are exactly within the special case of this soliton family, occurring in the presence of negative quartic-only dispersion. Here, we report on the first experimental generation of GQKSs from a fiber laser and investigate their temporal and spectral characteristics. We find that the temporal and spectral features of GQKSs are closely related to the quadratic dispersion as well as the pump power and intra-cavity birefringence. The flatness and the sideband numbers of the output spectra depends on the pump power and the intra-cavity phase delay bias under appropriate quadratic and quartic dispersions. GQKSs of different states, including harmonic mode-locking, the multiple solitons and soliton bunches, are also observed with proper engineering of the quadratic and quartic dispersion values. The formation mechanisms behind these different soliton states are illustrated. These results can facilitate the further understanding of unique features and evolution dynamics of GQKSs produced directly from ultrafast fiber lasers and accelerate their applications.