The optically driven acoustic modes and nonlinear response of plasmonic nanoparticles are important in many applications, but are strongly resonant, which restricts their excitation to predefined wavelengths. Here, we demonstrate that multilayered spherical plasmonic hetero-nanoparticles, formed by alternating layers of gold and silica, provide a platform for a broadband nonlinear optical response from visible to near-infrared wavelengths. They also act as a tunable optomechanical system with mechanically decoupled layers in which different acoustic modes can be selectively switched on/off by tuning the excitation wavelength. These observations not only expand the knowledge about the internal structure of composite plasmonic nanoparticles but also allow for an additional degree of freedom for controlling their nonlinear optical and mechanical properties.