We cross-correlate the largest available mid-infrared (Wide-field Infrared Survey Explorer– WISE), X-ray (3XMM) and radio (Faint Images of the Radio Sky at Twenty centimetres+NRAO VLA Sky Survey) catalogues to define the MIXR sample of AGN and star-forming galaxies. We pre-classify the sources based on their positions on the WISE colour/colour plot, showing that the MIXR triple selection is extremely effective to diagnose the star formation and AGN activity of individual populations, even on a flux/magnitude basis, extending the diagnostics to objects with luminosities and redshifts from SDSS DR12. We recover the radio/mid-IR star formation correlation with great accuracy, and use it to classify our sources, based on their activity, as radio-loud and radio-quiet active galactic nuclei (AGN), low excitation radio galaxies/low ionization nuclear emission line regions, and non-AGN galaxies. These diagnostics can prove extremely useful for large AGN and galaxy samples, and help develop ways to efficiently triage sources when data from the next generation of instruments becomes available. We study bias in detail, and show that while the widely used WISE colour selections for AGN are very successful at cleanly selecting samples of luminous AGN, they miss or misclassify a substantial fraction of AGN at lower luminosities and/or higher redshifts. MIXR also allows us to test the relation between radiative and kinetic (jet) power in radio-loud AGN, for which a tight correlation is expected due to a mutual dependence on accretion. Our results highlight that long-term AGN variability, jet regulation, and other factors affecting the Q/Lbol relation, are introducing a vast amount of scatter in this relation, with dramatic potential consequences on our current understanding of AGN feedback and its effect on star formation.