The quantum control of mechanical resonators requires the realization of exceptionally low dissipation in conjunction with strong nonlinear interactions. In this thesis, we demonstrate the simultaneous realization of both these features in ultrahigh quality factor silicon nitride membrane resonators, a promising new optomechanical platform in the emerging field of cavity optomechanics. The mechanical properties of the silicon nitride resonators are studied through a combination of spectroscopic and interferometric imaging techniques. We demonstrate ultrahigh quality factors of 5 x 107 and frequency-quality factor products of 1 x 1014 Hz corresponding to the largest values yet reported for mesoscopic membrane resonators, and an order of magnitude larger than what is required for room temperature quantum control. We perform a study of the limiting dissipation mechanisms as a function of resonator and substrate geometries and identify radiation loss through the supporting substrate as the dominant loss process. We proceed to alleviate radiation loss through the engineering of substrates with phononic bandgaps and present preliminary demonstrations of increased quality factors for a wide range of membrane modes. We also realize a two-mode parametric nonlinear process, and use it to demonstrate nondegenerate mechanical parametric amplification and two-mode thermomechanical noise squeezing. The observed phenomena show excellent agreement, over five orders of magnitude in displacement, with a two-mode model with the parametric coupling between the modes mediated by an excitation of the supporting substrate platform. The realization of a strong nonlinear interaction in a mechanical platform that is compatible with optomechanical cooling, room temperature quantum control and quantum limited detection is an important step towards the realization of non-classical mechanical states, the observation of entanglement between macroscopic mechanical degrees of freedom and quantum enhanced metrology. The last chapter of this thesis is unrelated to the rest of the work presented in it. This chapter discusses the coupling between the underlying elastic medium and electronic nematic order in high temperature cuprate superconductors and suggests the use of acoustic phonons as a probe of nematic order. [1] Dissipation in Ultrahigh Quality Factor SiN Membrane Resonators, S. Chakram, Y.S. Patil, L. Chang, and M. Vengalattore, Phys. Rev. Lett. 112, 127201, (2014). [2] Thermomechanical two-mode squeezing in an ultrahigh Q membrane resonator, Y. S. Patil, S. Chakram, L. Chang and M. Vengalattore, arXiv:1410.7109, (2014). [3] Multimode phononic correlations in a nondegenerate parametric amplifier, S. Chakram, Y. S. Patil, M. Vengalattore, arXiv:1412.8536, (2014). [4] Nondestructive imaging of an ultracold lattice gas, Y. S. Patil, S. Chakram, L. M. Aycock, and M. Vengalattore, Phys. Rev. A 90, 033422, (2014). [5] Quantum Control by Imaging : The Zeno effect in an ultracold lattice gas, Y. S. Patil, S. Chakram, M. Vengalattore, arXiv:1411.2678, (2014).