We bring together ac susceptibility and dc magnetization to uncover the rich magnetic field-temperature behavior of a series of rare earth indium oxides, RInO3 (R = Tb, Dy, and Gd). The degree of frustration is much larger than expected, particularly in TbInO3, and the ground states are glasslike with antiferromagnetic tendencies. The activation energy for spin reorientation is low. Chemical substitution with Mn3+ ions to form TbIn1- xMn xO3 ( x ≤ 0.01) relieves much of the frustration that characterizes the parent compound and slightly enhances the short-range antiferromagnetic order. The phase diagrams developed from this work reveal the rich competition between spin orders and provide an opportunity to compare the dynamics in the RInO3 and Mn-substituted systems. These structure-property relations may be useful for understanding magnetism in other geometrically frustrated multiferroics.