The Complete Local-Volume Groups Sample (CLoGS) was created in response to the lack of unbiased galaxy group samples and is designed to avoid the selection biases generally present particularly in X-ray selected samples (strongly biased in favor of the X-ray bright, centrally-concentrated cool-core systems). This statistically-complete sample of 53 groups within 80 Mpc is intended to serve as a representative survey of groups in the local Universe. In addition to X-ray data from Chandra and XMM (100% complete at this point, using both archival and new observations), we have added GMRT radio continuum observations (at 235 and 610 MHz, complete for the entire sample) and IRAM 30 m and APEX telescope observations of CO(1-0) and CO(2-1) lines (complete for the group-dominant early-type galaxies in the sample). We find that 14 of the 26 high-richness groups are X-ray bright, and that ∼53-65 per cent of the X-ray bright groups have cool cores, a somewhat lower fraction than found by previous archival surveys. Approximately 30 per cent of the X-ray bright groups show evidence of recent dynamical interactions (mergers or sloshing), and ∼35 per cent of their dominant early-type galaxies host active galactic nuclei with radio jets. In the 26 high-richness groups, 92% of the dominant galaxies host detected radio sources, with a four order-of-magnitude range in luminosities. Roughly half are point-like, with another quarter hosting jets and most of the rest showing a diffuse morphology. Jet sources are more common in X-ray bright groups, with radio non-detections found only in X-ray faint systems. We find that central AGN are not always in balance with cooling, but may instead produce powerful periodic bursts of feedback heating. Of the 53 CLoGS dominant galaxies, 21 are detected in CO and we confirm our previous findings that they have low star formation rates (0.01-1 Msun/yr) but short depletion times (<1 Gyr) implying rapid replenishment of their gas reservoirs. A much higher fraction of our group-dominant galaxies (~60%) are AGN-dominated than is the case for the general population of ellipticals.