We analyzed broadband X-ray and radio data of the magnetar SGR J1935+2154 taken in the aftermath of its 2014, 2015, and 2016 outbursts. The source soft X-ray spectrum <10 keV is well described with a blackbody+power-law (BB+PL) or 2BB model during all three outbursts. Nuclear Spectroscopic Telescope Array observations revealed a hard X-ray tail, with a PL photon index Γ = 0.9, extending up to 50 keV, with flux comparable to the one detected <10 keV. Imaging analysis of Chandra data did not reveal small-scale extended emission around the source. Following the outbursts, the total 0.5–10 keV flux from SGR J1935+2154 increased in concordance to its bursting activity, with the flux at activation onset increasing by a factor of ~7 following its strongest 2016 June outburst. A Swift/X-Ray Telescope observation taken 1.5 days prior to the onset of this outburst showed a flux level consistent with quiescence. We show that the flux increase is due to the PL or hot BB component, which increased by a factor of 25 compared to quiescence, while the cold BB component kT = 0.47 keV remained more or less constant. The 2014 and 2015 outbursts decayed quasi-exponentially with timescales of ~40 days, while the stronger 2016 May and June outbursts showed a quick short-term decay with timescales of about four days. Our Arecibo radio observations set the deepest limits on the radio emission from a magnetar, with a maximum flux density limit of 14 μJy for the 4.6 GHz observations and 7 μJy for the 1.4 GHz observations. We discuss these results in the framework of the current magnetar theoretical models.