We present densely sampled BVRI light curves of the optical transient associated with the gamma-ray burst (GRB) 030329, the result of a coordinated observing campaign conducted at five observatories. Augmented with published observations of this GRB, the compiled optical data set contains 2687 photometric measurements, obtained between 78 minutes and 79 days after the burst. This data set allows us to follow the photometric evolution of the transient with unprecedented detail. We use the data to constrain the light curve of the underlying supernova (SN) 2003dh and show that it evolved faster than and was probably somewhat fainter than the Type Ic SN 1998bw, associated with GRB 980425. We find that our data can be described by a broken power-law decay perturbed by a complex variable component. The early- and late-time decay slopes are determined to be α$_1$åisebox-0.5ex 1.1 and α$_2$i̊sebox-0.5ex 2. Assuming this single-break power-law model, we constrain the break to lie between rs̊ebox-0.5ex 3 and rae̊box-0.5ex 8 days after the burst. This simple, singly broken power-law model, derived only from the analysis of our optical observations, may also account for available multiband data, provided that the break happened raib̊ox-0.5ex 8 days after the burst. The more complex double-jet model of Berger et al. provides a comparable fit to the optical, X-ray, millimeter, and radio observations of this event. The unique early coverage available for this event allows us to trace the color evolution of the afterglow during the first hours after the burst. We detect a significant change in optical colors during the first day. Our color analysis is consistent with a cooling-break frequency sweeping through the optical band during the first day. The light curves of GRB 030329 reveal a rich array of variations, superposed over the mean power-law decay. We find that the early variations (<raiso̊x-0.5ex 8 days after the burst) are asymmetric, with a steep rise followed by a relatively slower (by a factor of about 2) decline. The variations maintain a similar timescale during the first 4 days and then get significantly longer. The structure of these variations is similar to those previously detected in the afterglows of several GRBs.