Position measurements of mechanical oscillators underpin experiments spanning from applied nanoscale sensing to endeavors aiming to resolve open fundamental problems of modern physics. Sufficiently precise position measurements are also used for engineering quantum states. Because of the coupling to the thermal environment, mechanical oscillators undergo Brownian motion, which sets the lowest measurement error when an oscillator is used as a probe and puts an upper bound on the decoherence time of mechanical quantum states. The thermal noise from the environment that drives the Brownian motion is proportional to the energy dissipation rate, and for this reason, mechanical resonators with low dissipation are the subject of broad and long-standing interest. The dissipation of vibrational modes in mechanical resonators is ultimately limited by intrinsic losses, which appear to be unavoidable as the acoustic strain creating them is also the source of potential energy for the vibrational modes. Intrinsic dissipation is nearly constant for different modes of bulk resonators. In striking contrast, the flexural modes of high aspect-ratio resonators subjected to static stress can experience very small, ``diluted", intrinsic dissipation. Understanding and engineering dissipation dilution led to remarkable progress in the development of low-loss mechanical resonators, which is in part covered in this thesis. Presently, chip-scale MHz-frequency mechanical resonators made of stoichiometric silicon nitride films are among the highest quality factor resonators that exist, reaching quality factors close to one billion at room temperature. The demonstration of these devices paves the way for force measurements with unprecedented resolution and the generation of long-lived quantum states of macroscopic objects. This thesis explores dissipation dilution in thin-film mechanical resonators and experimental position measurements performed on such resonators integrated into optomechanical cavities. Using an on-chip integrated optomechanical transducer, we implement the variational measurement strategy and demonstrate quantum correlations arising between the quadratures of meter light as a result of quantum measurement backaction. In our experiments at cryogenic temperature, these correlations lead to ponderomotive squeezing of light. The potential of the recently emerged record low-loss mechanical devices is particularly enticing at room temperature, as dissipation dilution is one of the few means to counteract the high thermal occupation of the bath modes. The large amplitudes of Brownian motion at room temperature, however, put a stringent limitation on the linearity range of the detector. As shown in this thesis, the interferometric nonlinearity inherent to optical measurements can easily become the dominant source of extrinsic thermal noise in detection.