  We report the transitions among different oscillation quenching states
induced by the interplay of diffusive (direct) coupling and environmental
(indirect) coupling in coupled identical oscillators. This coupling scheme was
introduced by Resmi {\it et al}. [Phys. Rev. E, 84, 046212 (2011)] as a general
scheme to induce amplitude death (AD) in nonlinear oscillators. Using a
detailed bifurcation analysis we show that in addition to AD, which actually
occurs only in a small region of parameter space, this coupling scheme can
induce other oscillation quenching states, namely oscillation death (OD) and a
novel nontrvial AD (NAD) state, which is a nonzero {\it bistable} homogeneous
steady state; more importantly, this coupling mediates a transition from AD to
OD state and a new transition from AD to NAD state. We identify diverse routes
to the NAD state and map all the transition scenarios in the parameter space
for periodic oscillators. Finally, we present the first experimental evidence
of oscillation quenching states and their transitions induced by the interplay
of direct and indirect coupling.
