Excitons in monolayer transition metal dichalcogenides (TMDs) provide a paradigm of composite Boson in a two-dimensional system. This Letter reports a photoluminescence and reflectance study of excitons in monolayer molybdenum diselenide (MoSe2) with electrostatic gating. We observe the repulsive and attractive Fermi polaron modes of the band edge exciton, its excited state, and the spin-off excitons, which the simple three-particle trion model is insufficient to explain. The contrasting energy shift between the exciton and charge-bound excitons (repulsive and attractive polaron modes) and the remarkably different gate dependence of the polaron energy splitting between the ground state and the excited state excitons unambiguously support the Fermi polaron picture for excitons in monolayer TMDs.