Both FeH and FeH+ are predicted to be abundant in cool stellar atmospheres and proposed to be molecular components of the gas phase interstellar medium (ISM). However, experimental and simulated data for both species are lacking, which have hindered astronomical detection. There are no published laboratory data for the spectroscopy of FeH+ in any frequency regime. It is also not established if FeH+ possesses salient multireference character, which would pose significant challenges for ab initio modeling of geometric and spectroscopic properties. With a set of high-level coupled cluster and multireference configuration interaction computations, a study of the electronic structure of the ground state and seven excited states of FeH+ was carried out. An X5Δi electronic ground state of FeH+ is found, in agreement with previous theoretical studies. Including corrections for spin-orbit coupling and anharmonic vibrational effects, the Ω = 3, ν = 0 spin ladder of the A5Πi electronic state lies 872 cm-1 higher in energy than the Ω = 4, ν = 0 spin ladder of the ground state. Combined with previous work in our laboratory, the ionization energy of FeH is computed to be 7.4851 eV. With modern multireference configuration interaction and coupled cluster methods, spectroscopic constants (re, Be, ωe, ωexe, αe, and D¯e) for several bound excited states (A5Πi, B 5Σi +, a 3Σr -, b3Φi, c3Πi, d3Δr, and 7Σ+) were characterized. This study will lead efforts to identify FeH+ in the ISM and help solve important remaining questions in quantifying metal-hydride bonding.