Microelectromechanical Systems are ubiquitous in modern technology, with applications ranging from accelerometers in smartphones to ultra-high precision motion stages used for atomically-precise positioning. With the appropriate selection of materials and device design, MEMS resonators with ultra-high quality factors can be fabricated at minimal cost. As the sizes of such resonators decrease, however, their mechanical, electrical, and material properties can no longer be treated as linear, as can be done for larger-scale devices. Unfortunately, adding nonlinear effects to a system changes its dynamics from exactly-solvable to only solvable in specific cases, if at all. Despite (and because of) these added complications, nonlinear effects open up an entirely new world of behaviors that can be measured or taken advantage of to create even more advanced technologies. In our resonators, oscillations are induced and measured using aluminum nitride transducers. I used this mechanism for several separate highly-sensitive experiments. In the first, I demonstrate the incredible sensitivity of these resonators by actuating a mechanical resonant mode using only the force generated by the radiation pressure of a laser at room temperature. In the following three experiments, which use similar mechanisms, I demonstrate information transfer and force measurements by taking advantage of the nonlinear behavior of the resonators. When nonlinear resonators are strongly driven, they exhibit sum and difference frequency generation, in which a large carrier signal can be mixed with a much smaller modulation to produce signals at sum and difference frequencies of the two signals. These sum and difference signals are used to detect information encoded in the modulation signal using optical radiation pressure and acoustic pressure waves. Finally, in my experiments, I probe the nonlinear nature of the piezoelectric material rather than take advantage of the nonlinear resonator behavior. The relative sizes of the linear and nonlinear portions of the piezoelectric constant can be determined because the force applied to the resonator by a transducer is independent of the dielectric constant. This method allowed me to quantify the nonlinear constants.