Acoustic-wave devices provide a promising chip-scale platform for efficiently coupling radio frequency (rf) and optical fields. Here, we use an integrated piezo-optomechanical circuit platform that exploits both the piezoelectric and photoelastic coupling mechanisms to link 2.4-GHz rf waves to 194-THz (1550 nm) optical waves, through coupling to propagating and localized 2.4-GHz acoustic waves. We demonstrate acousto-optic modulation, resonant in both the optical and mechanical domains, in which waveforms encoded on the rf carrier are mapped to the optical field. We also show optoacoustic gating, in which the application of modulated optical pulses interferometrically gates the transmission of propagating acoustic pulses. The time-domain characteristics of this system under both pulsed rf and pulsed optical excitation are considered in the context of the different physical pathways involved in driving the acoustic waves, and modeled through the coupled mode equations of cavity optomechanics.