SGR 1833−0832 was discovered on 2010 March 19, thanks to the Swift detection of a short hard X-ray burst and follow-up X-ray observations. Since then, it was repeatedly observed with Swift, Rossi X-ray Timing Explorer and XMM-Newton. Using these data, which span about 225 d, we studied the long-term spectral and timing characteristics of SGR 1833−0832. We found evidence for diffuse emission surrounding SGR 1833−0832, which is most likely a halo produced by the scattering of the point-source X-ray radiation by dust along the line of sight, and we show that the source X-ray spectrum is well described by an absorbed blackbody, with temperature kT∼ 1.2 keV and absorbing column N H= (10.4 ± 0.2) × 1022 cm−2, while different or more complex models are disfavoured. The source persistent X-ray emission remained fairly constant at ∼3.7 × 10−12 erg cm−2 s−1 for the first ∼20 d after the onset of the bursting episode, then it faded by a factor of ∼40 in the subsequent ∼140 d, following a power-law trend with index α≃−0.5. We obtained a phase-coherent timing solution with the longest baseline (∼225 d) to date for this source which, besides period P= 7.565 4084(4) s and period derivative s s−1, includes higher order period derivatives. We also report on our search of the counterpart to the soft gamma-ray repeater (SGR) at radio frequencies using the Australia Telescope Compact Array and the Parkes Radio Telescope. No evidence for radio emission was found, down to flux densities of 0.9 mJy (at 1.5 GHz) and 0.09 mJy (at 1.4 GHz) for the continuum and pulsed emissions, respectively, consistently with other observations at different epochs. Finally, the analysis of the field of PSR B1830−08 (J1833−0827), which was serendipitously imaged by the XMM-Newton observations, led to the discovery of the X-ray pulsar wind nebula generated by this 85-ms radio pulsar. We discuss its possible association with the unidentified TeV source HESS J1834−087.