The ATLAS Inner Detector is a composite tracking system consisting of silicon pixels, silicon strips and straw tubes in a 2 T magnetic field. Its installation was completed in August 2008 and the detector took part in data-taking with single LHC beams and cosmic rays. The initial detector operation, hardware commissioning and in-situ calibrations are described. Tracking performance has been measured with 7.6 million cosmic-ray events, collected using a tracking trigger and reconstructed with modular pattern-recognition and fitting software. The intrinsic hit efficiency and tracking trigger efficiencies are close to 100%. Lorentz angle measurements for both electrons and holes, specific energy-loss calibration and transition radiation turn-on measurements have been performed. Different alignment techniques have been used to reconstruct the detector geometry. After the initial alignment, a transverse impact parameter resolution of 22.1 +/- 0.9 mu m and a relative momentum resolution sigma (p) /p=(4.83 +/- 0.16)x10(-4) GeV(-1)xp (T) have been measured for high momentum tracks.