This Thesis is based on four original publications and concludes my theoretical research during the years of my Ph.D. studies. Firstly we proposed a new way to deterministically transfer an arbitrary quantum state of light to the mechanical oscillator. It is shown that it is possible to enhance the coupling of light to matter with the help of only local Gaussian operation on the light. This approach is proved to help to transfer negativity of Winger function from light to mechanics. Next, we introduced a scheme efficiently entangling two distant mechanical oscillators mediated by the optical or microwave field. At the time the work was performed, there have been no mechanical oscillators coupled at the quantum level demonstrated. The proposed scheme assumes a certain coupling between the mechanical systems -- the quantum nondemolition (QND) one, which is very useful for basic continuous-variable quantum gates. We also studied quantum transducers which are very important for the development of quantum technology. The proposed transducer is based on a sequence of long-pulsed interactions between the systems of interest (optical or microwave fields) and the mediating system (mechanical oscillator). We showed that with the help of the geometric phase effect it is possible to eliminate the noisy influence of the mechanical mediator. To follow the development of quantum optomechanics, we explored the very similar transducer, but in the regime of high-intensive ultra-short pulses (stroboscopic regime). It was unclear before whether the geometric phase effect will be sufficient to obtain a robust transducer in this regime. Our proposal is suitable for arbitrary wavelengths of radiation which might stimulate a much broader class of feasible quantum transducers mediated by mechanical systems. We believe that this thesis supported research and advanced the field making important theoretical explorations that open the way for future experimental implementations and studies of other setups based on similar principles.This Thesis is based on four original publications and concludes my theoretical research during the years of my Ph.D. studies. Firstly we proposed a new way to deterministically transfer an arbitrary quantum state of light to the mechanical oscillator. It is shown that it is possible to enhance the coupling of light to matter with the help of only local Gaussian operation on the light. This approach is proved to help to transfer negativity of Winger function from light to mechanics. Next, we introduced a scheme efficiently entangling two distant mechanical oscillators mediated by the optical or microwave field. At the time the work was performed, there have been no mechanical oscillators coupled at the quantum level demonstrated. The proposed scheme assumes a certain coupling between the mechanical systems -- the quantum nondemolition (QND) one, which is very useful for basic continuous-variable quantum gates. We also studied quantum transducers which are very important for the development of quantum technology. The proposed transducer is based on a sequence of long-pulsed interactions between the systems of interest (optical or microwave fields) and the mediating system (mechanical oscillator). We showed that with the help of the geometric phase effect it is possible to eliminate the noisy influence of the mechanical mediator. To follow the development of quantum optomechanics, we explored the very similar transducer, but in the regime of high-intensive ultra-short pulses (stroboscopic regime). It was unclear before whether the geometric phase effect will be sufficient to obtain a robust transducer in this regime. Our proposal is suitable for arbitrary wavelengths of radiation which might stimulate a much broader class of feasible quantum transducers mediated by mechanical systems. We believe that this thesis supported research and advanced the field making important theoretical explorations that open the way for future experimental implementations and studies of other setups based on similar principles.