This series of two papers reports the investigation of the properties of multiply charged gold clusters cations Aunq+ (n = 3 -20) up to charge q = 4. In the present part I of the study, a global exploration of their potential energy surface has been performed using a combination of parallel tempering molecular dynamics and quenches performed at the DFTB level. When increasing the charge of the clusters, their structure was found to evolve from a compact form to an elongated one, shifting the 2D/3D transition toward larger sizes. Such structure elongation is explained by the minimization of coulombic destabilization. In the further part II of this study, the stability of these low-energy isomers will be discussed, with a focus on the cluster's ionization and fragmentation.