Solar soft X-ray (SXR) observations provide important diagnostics of plasma heating, during solar flares and quiescent times. Spectrally- and temporally-resolved measurements are crucial for understanding the dynamics and evolution of these energetic processes; spatially-resolved measurements are critical for understanding energy transport. A better understanding of the thermal plasma informs our interpretation of hard X-ray (HXR) observations of nonthermal particles, improving our understanding of the relationships between particle acceleration, plasma heating, and the underlying release of magnetic energy during reconnection. We introduce a new proposed mission, the CubeSat Imaging X-ray Solar Spectrometer (CubIXSS), to measure spectrally- and spatially-resolved SXRs from the quiescent and flaring Sun from a 6U CubeSat platform in low-Earth orbit during a nominal 1-year mission. CubIXSS includes the Amptek X123-SDD silicon drift detector, a low-noise, commercial off-the-shelf (COTS) instrument enabling solar SXR spectroscopy from ~0.5 to ~30 keV with ~0.15 keV FWHM spectral resolution with low power, mass, and volume requirements. An X123-CdTe cadmium-telluride detector is also included for ~5-100 keV HXR spectroscopy with ~0.5-1 keV FWHM resolution. CubIXSS also includes a novel spectro-spatial imager -- the first ever solar imager on a CubeSat -- utilizing a pinhole aperture and X-ray transmission diffraction grating to provide full-Sun imaging from ~0.1 to ~10 keV, with ~25 arcsec and ~0.1 Å FWHM spatial and spectral resolutions, respectively. We discuss scaled versions of these instruments, with greater sensitivity and dynamic range, and significantly improved spectral and spatial resolutions for the imager, for deployment on larger platforms such as Small Explorer missions.