We present an experimental study of the two-dimensional S=1/2 square-lattice antiferromagnet Cu(pz)(2)(ClO4)(2) (pz denotes pyrazine-C4H4N2) using specific-heat measurements, neutron diffraction, and cold-neutron spectroscopy. The magnetic field dependence of the magnetic ordering temperature was determined from specific-heat measurements for fields perpendicular and parallel to the square-lattice planes, showing identical field-temperature phase diagrams. This suggest that spin anisotropies in Cu(pz)(2)(ClO4)(2) are small. The ordered antiferromagnetic structure is a collinear arrangement with the magnetic moments along either the crystallographic b or c axis. The estimated ordered magnetic moment at zero field is m(0)=0.47 (5)mu(B) and thus much smaller than the available single-ion magnetic moment. This is evidence for strong quantum fluctuations in the ordered magnetic phase of Cu(pz)(2)(ClO4)(2). Magnetic fields applied perpendicular to the square-lattice planes lead to an increase in the antiferromagnetically ordered moment to m(0)=0.93 (5)mu(B) at mu H-0= 13.5 T evidence that magnetic fields quench quantum fluctuations. Neutron spectroscopy reveals the presence of a gapped spin excitations at the antiferromagnetic zone center and it can be explained with a slightly anisotropic nearest-neighbor exchange coupling described by J(xy)(1)= 1.563 (13) meV and J(z)(1) = 0.9979(2)J(1)(xy).