GRB 190829A at z = 0.0785 is the fourth closest long GRB ever detected by the Neil Gehrels Swift observatory, and the third confirmed case with a very high-energy component. We present our multiwavelength analysis of this rare event, focusing on its early stages of evolution, and including data from Swift, the MASTER global network of optical telescopes, ALMA, and ATCA. We report sensitive limits on the linear polarization of the optical emission, disfavouring models of off-axis jets to explain the delayed afterglow peak. The study of the multiwavelength light curves and broad-band spectra supports a model with at least two emission components: a bright reverse shock emission, visible at early times in the optical and X-rays and, later, in the radio band; and a forward shock component dominating at later times and lower radio frequencies. A combined study of the prompt and afterglow properties shows many similarities with cosmological long GRBs, suggesting that GRB 190829A is an example of classical GRBs in the nearby universe.