This work developed a multi-scale model, combining the macro-scale model for heat transfer and fluid flow with the micro-scale phase-field model for polycrystalline alloy solidification, to investigate the grain/sub-grain structure evolution during solidification in laser welding of 5083 Al sheets. Special attentions were paid to the planar to cellular and dendritic growth. Results demonstrated that, during planar-to-cellular transition, the initial planar instability primarily resulted from the rapid change in growth rate. The surface tension anisotropy can significantly affect the initial planar instability. With the increase of misorientation angle, the interfacial stiffness tended to increase, which enhanced the stabilizing effects of surface tension and thus made the S/L interface more and more stable. Furthermore, Results elucidated how solidification morphology develops during cellular-to-dendritic transition. Two important phenomena, namely, tertiary branching and tip-splitting, which were responsible for the decrease in primary dendritic arm spacing, were discovered. Finally, results illustrated how multiple grains with various orientations grow in competition in laser weld pool. Two competitive mechanisms of grain growth were revealed, namely, lateral expansion of favorably-oriented (FO) grain and blocking its unfavorably-oriented (UO) grain neighbors, and the elimination of UO grain by FO grain at the converging grain boundary.