We report on gamma-ray observations of the Crab Pulsar and Nebula using 8 months of survey data with the Fermi Large Area Telescope (LAT). The high quality light curve obtained using the ephemeris provided by the Nancay and Jodrell Bank radio telescopes shows two main peaks stable in phase with energy. The first gamma-ray peak leads the radio main pulse by (281 \pm 12 \pm 21) mus, giving new constraints on the production site of non-thermal emission in pulsar magnetospheres. The improved sensitivity and the unprecedented statistics afforded by the LAT enable precise measurement of the Crab Pulsar spectral parameters: cut-off energy at E_c = (5.8 \pm 0.5 \pm 1.2) GeV, spectral index of Gamma = (1.97 \pm 0.02 \pm 0.06) and integral photon flux above 100 MeV of (2.09 \pm 0.03 \pm 0.18) x 10^{-6} cm^{-2} s^{-1}. The first errors represent the statistical error on the fit parameters, while the second ones are the systematic uncertainties. Pulsed gamma-ray photons are observed up to ~ 20 GeV which precludes emission near the stellar surface, below altitudes of around 4 to 5 stellar radii in phase intervals encompassing the two main peaks. The spectrum of the nebula in the energy range 100 MeV - 300 GeV is well described by the sum of two power-laws of indices Gamma_{sync} = (3.99 \pm 0.12 \pm 0.08) and Gamma_{IC} = (1.64 \pm 0.05 \pm 0.07), corresponding to the falling edge of the synchrotron and the rising edge of the inverse Compton components, respectively. This latter, which links up naturally with the spectral data points of Cherenkov experiments, is well reproduced via inverse Compton scattering from standard Magnetohydrodynamics (MHD) nebula models, and does not require any additional radiation mechanism. Show Partial Abstract