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<jats:title>ABSTRACT</jats:title> <jats:p>We present the first three-frequency South Pole Telescope (SPT) cosmic microwave background (CMB) power spectra. The band powers presented here cover angular scales 2000 &lt; ℓ &lt; 9400 in frequency bands centered at 95, 150, and 220 GHz. At these frequencies and angular scales, a combination of the primary CMB anisotropy, thermal and kinetic Sunyaev–Zel'dovich (SZ) effects, radio galaxies, and cosmic infrared background (CIB) contributes to the signal. We combine <jats:italic>Planck</jats:italic>/HFI and SPT data at 220 GHz to constrain the amplitude and shape of the CIB power spectrum and find strong evidence for nonlinear clustering. We explore the SZ results using a variety of cosmological models for the CMB and CIB anisotropies and find them to be robust with one exception: allowing for spatial correlations between the thermal SZ effect and CIB significantly degrades the SZ constraints. Neglecting this potential correlation, we find the thermal SZ power at 150 GHz and ℓ = 3000 to be 3.65 ± 0.69 μK<jats:sup>2</jats:sup>, and set an upper limit on the kinetic SZ power to be less than 2.8 μK<jats:sup>2</jats:sup> at 95% confidence. When a correlation between the thermal SZ and CIB is allowed, we constrain a linear combination of thermal and kinetic SZ power: <jats:italic>D</jats:italic> <jats:sup>tSZ</jats:sup> <jats:sub>3000</jats:sub> + 0.5<jats:italic>D</jats:italic> <jats:sub>3000</jats:sub> <jats:sup>kSZ</jats:sup> = 4.60 ± 0.63 μK<jats:sup>2</jats:sup>, consistent with earlier measurements. We use the measured thermal SZ power and an analytic, thermal SZ model calibrated with simulations to determine σ<jats:sub>8</jats:sub> = 0.807 ± 0.016. Modeling uncertainties involving the astrophysics of the intracluster medium rather than the statistical uncertainty in the measured band powers are the dominant source of uncertainty on σ<jats:sub>8</jats:sub>. We also place an upper limit on the kinetic SZ power produced by patchy reionization; a companion paper uses these limits to constrain the reionization history of the universe.</jats:p>