The formation of a core collapse supernovae (SNe) results in a fast (but non- or mildly-relativistic) shock wave expanding outwards into the surrounding medium. The medium itself is likely modified due to the stellar mass-loss from the massive star progenitor, which may be Wolf-Rayet stars (for Type Ib/c SNe), red supergiant stars (for type IIP and perhaps IIb and IIL SNe), or some other stellar type. The wind mass-loss parameters determine the density structure of the surrounding medium. Combined with the velocity of the SN shock wave, this regulates the shock acceleration process. In this article we discuss the essential parameters that control the particle acceleration and gamma-ray emission in SNe, with particular reference to the Type IIb SN 1993J. The shock wave expanding into the high density medium leads to fast particle acceleration, giving rise to rapidly-growing plasma instabilities driven by the acceleration process itself. The instabilities grow over intraday timescales. This growth, combined with the interplay of non-linear processes, results in the amplification of the magnetic field at the shock front, which can adequately account for the magnetic field strengths deduced from radio monitoring of the source. The maximum particle energy can reach, and perhaps exceed, 1 PeV, depending on the dominant instability. The gamma-ray signal is found to be heavily absorbed by pair production process during the first week after the outburst. We derive the time dependent particle spectra and associated hadronic signatures of secondary particles (gamma-ray, leptons and neutrinos) arising from proton proton interactions. We find that the Cherenkov Telescope Array (CTA) should be able to detect objects like SN 1993J above 1 TeV. We predict a low neutrino flux above 10 TeV, implying a detectability horizon with current or planned neutrino telescopes of 1 Mpc.