The adsorption capacities of ion exchangers with the primary amine (Lewatit® VP OC 1065) and polyamine (Diaion™ CR20) functional groups relative to Pd(II) and Cu(II) ions were tested in a batch system, taking into account the influence of the acid concentration (HCl: 0.1–6 mol/L; HCl-HNO3: 0.9–0.1 mol/L HCl—0.1–0.9 mol/L HNO3), phase contact time (1–240 min), initial concentration (10–1000 mg/L), agitation speed (120–180 rpm), bead size (0.385–1.2 mm), and temperature (293–333 K), as well as in a column system where the variable operating parameters were HCl and HNO3 concentrations. There were used the pseudo-first order, pseudo-second order, and intraparticle diffusion models to describe the kinetic studies and the Langmuir and Freundlich isotherm models to describe the equilibrium data to obtain better knowledge about the adsorption mechanism. The physicochemical properties of the ion exchangers were characterized by the nitrogen adsorption/desorption analyses, CHNS analysis, Fourier transform infrared spectroscopy, the sieve analysis, and points of zero charge measurements. As it was found, Lewatit® VP OC 1065 exhibited a better ability to remove Pd(II) than Diaion™ CR20, and the adsorption ability series for heavy metals was as follows: Pd(II) >> Zn(II) ≈ Ni(II) >> Cu(II). The optimal experimental conditions for Pd(II) sorption were 0.1 mol/L HCl, agitation speed 180 rpm, temperature 293 K, and bead size fraction 0.43 mm ≤ f3 < 0.6 mm for Diaion™ CR20 and 0.315–1.25 mm for Lewatit® VP OC 1065. The maximum adsorption capacities were 289.68 mg/g for Lewatit® VP OC 1065 and 208.20 mg/g for Diaion™ CR20. The greatest adsorption ability of Lewatit® VP OC 1065 for Pd(II) was also demonstrated in the column studies. The working ion exchange in the 0.1 mol/L HCl system was 0.1050 g/mL, much higher compared to Diaion™ CR20 (0.0545 g/mL). The best desorption yields of %D1 = 23.77% for Diaion™ CR20 and 33.57% for Lewatit® VP OC 1065 were obtained using the 2 mol/L NH3·H2O solution.