Core-collapse supernovae have been supposed to be one of the most plausible sources of gravitational waves. Based on a series of our magnetohydrodynamic core-collapse simulations, we find that the gravitational amplitudes at core bounce can be within the detection limits for the currently running laser-interferometers for a galactic supernova if the central core rotates sufficiently rapidly. This is regardless of the difference of the realistic equations of state and the possible occurrence of the QCD phase transition near core bounce. Even if the core rotates slowly, we point out that the gravitational waves generated from anisotropic neutrino radiation in the postbounce phase due to the standing accretion shock instability (SASI) could be within the detection limits of the detectors in the next generation such as LCGT and the advanced LIGO for the galactic source. Since the waveforms significantly depend on the exploding scenarios, our results suggest that we can obtain the information about the long-veiled explosion mechanism from the gravitational wave signals. Furthermore we discuss the gravitational wave background (GWB) from the explosions of Pop III stars and show that the GWB from Pop III, depending on their formation rates, can be large enough to be within the detection limits of future planned interferometers such as DECIGO and BBO in the frequency interval of ~0.1-1 Hz. This means that the detections of GW background from Pop III stars can be an important tool to supply the information about the star formation history in the early universe.