Hybrid organic-inorganic perovskites have attractive optoelectronic properties including exceptional solar cell performance. The improved properties of perovskites have been attributed to polaronic effects involving stabilization of localized charge character by structural deformations and polarizations. Here we examine the Pb-I structural dynamics leading to polaron formation in methylammonium lead iodide perovskite by transient absorption, time-domain Raman spectroscopy, and density functional theory. Methylammonium lead iodide perovskite exhibits excited-state coherent nuclear wave packets oscillating at ~20, ~43, and ~75 cm^-1 which involve skeletal bending, in-plane bending, and c-axis stretching of the I-Pb-I bonds, respectively. The amplitudes of these wave packet motions report on the magnitude of the excited-state structural changes, in particular, the formation of a bent and elongated octahedral PbI₆^4- geometry. We have predicted the excited-state geometry and structural changes between the neutral and polaron states using a normal-mode projection method, which supports and rationalizes the experimental results. This study reveals the polaron formation via nuclear dynamics that may be important for efficient charge separation.