Rare-earth iron garnet films with perpendicular magnetic anisotropy could open new perspectives for spintronics. Holmium iron garnet (Ho₃Fe₅O₁₂, HoIG) films with thickness ranging from 2 to 100 nm are epitaxially grown on (111) orientated gadolinium gallium garnet single crystal substrate doped with yttrium and scandium (Gd_0.63Y_2.37Sc₂Ga₃O₁₂, GYSGG) by ultra-high vacuum magnetron sputtering. A 3-nm Pt film is further deposited on each of the HoIG films. The magnetic anisotropy and magneto-transport properties of heterostructures at room temperature are investigated. It is shown that the HoIG film as thin as 2 nm (less than two unit cells in thickness) exhibits the ferromagnetic properties at room temperature, and perpendicular magnetic anisotropy is achieved in the 2-60 nm thick films, and a maximum effective perpendicular anisotropy field reaches 350 mT due to the strain induced magnetoelastic anisotropy. The HoIG/Pt heterostructure shows significant anomalous Hall effect (AHE) and appreciable spin-Hall magnetoresistance (SMR) and/or anisotropic magnetoresistance (AMR). Remarkably, the AHE starts to decline gradually when the HoIG thickness is less than 4 nm, but the magnetoresistance decreases rapidly with the HoIG layer becoming less than 7 nm in thickness. The fact that the AHE in the heterostructure is less sensitive to the HoIG thickness suggests that the interface effect is more dominant in the AHE mechanism, whereas the bulk magnetic properties of the HoIG plays a more important role for the observed magnetoresistance. In addition, the spin Seebeck effect decreases exponentially with the decrease of HoIG thickness till the ultrathin limit, which was previously validated in the micrometer-thick YIG/Pt stacks in the frame of thermally excited magnon accumulation and propagation. The present results show that the nanometer HoIG/Pt heterostructure with tunable perpendicular magnetic anisotropy and efficient interfacial spin exchange interaction could be a promising candidate for insulating magnet based spintronic devices.