The Kardar-Parisi-Zhang equation for surface growth is analyzed in the regime where the nonlinear coupling constant is small. We present detailed calculations for the mean-square surface width in terms of the bare parameters of the equation. For surface dimension d≤2, this quantity is shown to obey crossover scaling. The case d=2 is marked by an exponentially slow crossover associated with the marginally unstable character of the linear theory. For d>2 a renormalization-group analysis in the one-loop approximation yields a logarithmic scaling form at the roughening transition between smooth and rough growth phases. The crossover behavior on either side of this transition is discussed.