The space between stars and galaxies is permeated by diffuse plasma. This plasma induces dispersion and scattering of astrophysical radio transients, observations of which can constrain the plasma density and its fluctuations. This dissertation combines in situ and remote probes, including the Voyager Interstellar Mission, pulsars, and extragalactic fast radio bursts (FRBs), to characterize plasma densities across a wide range of astrophysical environments: the very local interstellar medium (VLISM) near the heliosphere, the local ISM and circumgalactic medium (CGM) of the Milky Way, and the intergalactic medium (IGM) and host galaxies of FRBs. The detection of a persistent plasma wave signal in Voyager 1 data is used to regularly sample the very local ISM at sub-au resolution, yielding an electron density fluctuation spectrum that is compared to density fluctuations inferred from pulsar bow shocks and dispersion measure variations. Pulsar dispersion measures and parallax distances are used to constrain the scale-height, mid-plane density, and clumpiness of the Galactic warm ionized medium (WIM). FRB scattering and dispersion measurements are combined with pulsar observations to characterize the strength of density fluctuations in the CGM. The large dispersion and scattering of FRB 20190520B are used to infer host galaxy ISM properties, and the first detection of scattering variations from an FRB's circumsource medium is reported. Simulations of scattering and dispersion extending from the Milky Way to potential FRB host galaxies at z~5 reveal that a significant fraction of high-redshift FRBs may be undetected due to scattering, and prospects for constraining the multi-phase CGM and tenuous IGM with FRBs are assessed.