We present an investigation of the supernova remnant (SNR) G306.3-0.9, using archival multiwavelength data. The Suzaku spectra are well described by two-component thermal plasma models: the soft component is in ionization equilibrium and has a temperature similar to 0.59 keV, while the hard component has temperature similar to 3.2 keV and ionization time-scale similar to 2.6 x 10(10) cm(-3) s. We clearly detected Fe K-shell line at energy of similar to 6.5 keV from this remnant. The overabundances of Si, S, Ar, Ca and Fe confirm that the X-ray emission has an ejecta origin. The centroid energy of the Fe K line supports that G306.3-0.9 is a remnant of a Type Ia supernova (SN) rather than a core-collapse SN. The GeV gamma-ray emission from G306.3-0.9 and its surrounding were analysed using about 6-yr of Fermi data. We report about the non-detection of G306.3-0.9 and the detection of a new extended gamma-ray source in the south-west of G306.3-0.9 with significance of similar to 13 sigma. We discuss several scenarios for these results with the help of data from other wavebands to understand the SNR and its neighbourhood.