We report a neutron scattering study on the tetragonal compound Sr2Cu3O4Cl2, which has two-dimensional (2D) interpenetrating CuI and CuII subsystems, each forming a S=1/2 square lattice quantum Heisenberg antiferromagnet (SLQHA). The mean-field ground state is degenerate, since the intersubsystem interactions are geometrically frustrated. Magnetic neutron scattering experiments show that quantum fluctuations lift the degeneracy and cause a 2D Ising ordering of the CuII subsystem. Due to quantum fluctuations a dramatic increase of the CuI out-of-plane spin-wave gap is also observed. The temperature dependence and the dispersion of the spin-wave energy are quantitatively explained by spin-wave calculations which include quantum fluctuations explicitly. The values for the nearest-neighbor superexchange interactions between the CuI and CuII ions and between the CuII ions are determined experimentally to be JI−II=−10(2) meV and JII=10.5(5) meV, respectively. Due to its small exchange interaction JII, the 2D dispersion of the CuII SLQHA can be measured over the whole Brillouin zone with thermal neutrons, and a dispersion at the zone boundary, predicted by theory, is confirmed. The instantaneous magnetic correlation length of the CuII SLQHA is obtained up to a very high temperature, T/JII≈0.75. This result is compared with several theoretical predictions as well as recent experiments on the S=1/2 SLQHA.