Advances in cooling and trapping neutral atoms enabled the investigation of ultracold fermions with an enlarged SU(N) spin symmetry, and offered unique possibilities for exploring correlated quantum many-body systems with high spin. In such a fermionic system of ultracold two-electron ytterbium, multi-component fermions interact identically with others owing to SU(N) symmetry with tunable N=1,...,6. In this talk, we report our progress toward a quantitative thermodynamic study of SU(N) fermions in the Fermi liquid regime, including experiments on measuring compressibility, revealing bosonization and investigating collective excitations. We highlight the highly tunable features of the system, such as spin multiplicity, spin imbalance and dimensions. These developments, together with recent experiments from other groups, set the stage for exploring SU(N) spin symmetry in the context of magnetic correlations both in bulk and lattices.