Bis(pyrazol-1-yl)acetic acid (HC(pz)2COOH) and bis(3,5-dimethyl-pyrazol-1-yl)acetic acid (HC(pzMe2)2COOH) were converted into the methyl ester derivatives 1 (LOMe) and 2 (L2OMe), respectively, and were used for the preparation of Cu(I) and Cu(II) complexes 3–10. The copper(II) complexes were prepared by the reaction of CuCl2·2H2O or CuBr2 with ligands 1 and 2 in methanol solution. The copper(I) complexes were prepared by the reaction of Cu[(CH3CN)4]PF6 and 1,3,5-triaza-7-phosphaadamantane (PTA) or triphenylphosphine with LOMe and L2OMe in acetonitrile solution. Synchrotron radiation-based complementary techniques (XPS, NEXAFS, and XAS) were used to investigate the electronic and molecular structures of the complexes and the local structure around copper ions in selected Cu(I) and Cu(II) coordination compounds. All Cu(I) and Cu(II) complexes showed a significant in vitro antitumor activity, proving to be more effective than the reference drug cisplatin in a panel of human cancer cell lines, and were able to overcome cisplatin resistance. Noticeably, Cu complexes appeared much more effective than cisplatin in 3D spheroid cultures. Mechanistic studies revealed that the antitumor potential did not correlate with cellular accumulation but was consistent with intracellular targeting of PDI, ER stress, and paraptotic cell death induction.