The investigation of Ricci solitons is the focus of this work. We have proved triviality results for compact gradient Ricci soliton under certain restriction. Later, a rigidity result is derived for a compact gradient shrinking Ricci soliton. Also, we have estimated the conjugate radius for non-compact gradient shrinking Ricci solitons with superharmonic potential. Moreover, an upper bound for the conjugate radius of Ricci soliton with concircular potential vector field is determined. Finally, it is proved that a non-compact gradient Ricci soliton with a pole and non-negative Ricci curvature is non-shrinking.