CUORE-0 is a cryogenic detector that uses an array of tellurium dioxide bolometers to search for neutrinoless double-beta decay of ¹³⁰Te. We present the first data analysis with 7.1 kg⋅yr of total TeO₂ exposure focusing on background measurements and energy resolution. The background rates in the neutrinoless double-beta decay region of interest (2.47 to 2.57 MeV) and in the α background-dominated region (2.70 to 3.90 MeV) have been measured to be 0.07 ± 0.011 and 0.019 ± 0.002 counts/(keV⋅kg⋅y), respectively. The latter result represents a factor of 6 improvement from a predecessor experiment, Cuoricino. The results verify our understanding of the background sources in CUORE-0, which is the basis of extrapolations to the full CUORE detector. The obtained energy resolution (full width at half maximum) in the region of interest is 5.7 keV. Based on the measured background rate and energy resolution in the region of interest, CUORE-0 half-life sensitivity is expected to surpass the observed lower bound of Cuoricino with one year of live time.