While pulsars possess exceptional rotational stability, large-scale timing studies have revealed at least two distinct types of irregularities in their rotation: red timing noise and glitches. Using modern Bayesian techniques, we investigated the timing noise properties of 300 bright southern-sky radio pulsars that have been observed over 1.0-4.8yr by the upgraded Molonglo Observatory Synthesis Telescope (MOST). We reanalysed the spin and spin-down changes associated with nine previously reported pulsar glitches, report the discovery of three new glitches and four unusual glitch-like events in the rotational evolution of PSR J1825-0935. We develop a refined Bayesian framework for determining how red noise strength scales with pulsar spin frequency ({nu}) and spin-down frequency (d{nu}/dt), which we apply to a sample of 280 non-recycled pulsars. With this new method and a simple power-law scaling relation, we show that red noise strength scales across the non-recycled pulsar population as {nu}^a^|d{nu}/dt|^b^, where a=-0.84^+0.47^_-0.49_ and b=0.97^+0.16^_-0.19_. This method can be easily adapted to utilize more complex, astrophysically motivated red noise models. Lastly, we highlight our timing of the double neutron star PSR J0737-3039, and the rediscovery of a bright radio pulsar originally found during the first Molonglo pulsar surveys with an incorrectly catalogued position.