Hysteresis properties and magnetization reversal in TbCo(30 nm) and FeNi(10 nm)/TbCo(30 nm) films with nanoscale antidot lattices are investigated to test the effect of nanoholes on the perpendicular anisotropy in the TbCo layer and the induced exchange bias in the FeNi layer. The antidots are introduced by depositing the films on top of hexagonally ordered porous anodic alumina substrates with pore diameter and interpore distance fixed to 75 nm and 105 nm, respectively. The analysis of combined vibrating sample magnetometry, Kerr microscopy and magnetic force microscopy imaging measurements has allowed us to link macroscopic and local magnetization reversal processes. For magnetically hard TbCo films, we demonstrate the tunability of magnetic anisotropy and coercive field (i.e., it increases from 0.2 T for the continuous film to 0.5 T for the antidot film). For the antidot FeNi/TbCo film, magnetization of FeNi is confirmed to be in plane. Although an exchange bias has been locally detected in the FeNi layer, the integrated hysteresis loop has increased coercivity and zero shift along the field axis due to the significantly decreased magnetic anisotropy of TbCo layer.