Close white dwarf binaries consisting of a white dwarf and an A-, F-, G-, or K-type main-sequence star, henceforth close WD+AFGK binaries, are ideal systems to understand the nature of type Ia supernovae progenitors and to test binary evolution models. In this work we identify 775 WD+AFGK candidates from TGAS (The Tycho-Gaia Astrometric Solution) and Gaia Data Release 2 (DR2), a well-defined sample of stars with available parallaxes, and we measure radial velocities (RVs) for 275 of them with the aim of identifying close binaries. The RVs have been measured from high-resolution spectra obtained at the Xinglong 2.16 m Telescope and the San Pedro Mártir 2.12 m Telescope and/or from available LAMOST DR6 (low-resolution) and RAVE DR5 (medium-resolution) spectra. We identify 23 WD+AFGK systems displaying more than 3 RV variation among 151 systems for which the measured values are obtained from different nights. Our WD+AFGK binary sample contains both AFGK dwarfs and giants, with a giant fraction ∼43%. The close binary fractions we determine for the WD+AFGK dwarf and giant samples are 24% and 15%, respectively. We also determine the stellar parameters (i.e., effective temperature, surface gravity, metallicity, mass, and radius) of the AFGK companions with available high-resolution spectra. The stellar parameter distributions of the AFGK companions that are members of close and wide binary candidates do not show statistically significant differences.