We perform a detailed statistical study of the distribution of topological and spectral indices on random graphs \( G=(V,E) \) in a wide range of connectivity regimes. First, we consider degree-based topological indices (TIs), and focus on two classes of them: \( X_\Sigma(G) = \sum_{uv \in E} f(d_u,d_v) \) and \( X_\Pi(G) = \prod_{uv \in E} g(d_u,d_v) \), where \( uv \) denotes the edge of \( G \) connecting the vertices \( u \) and \( v \), \( d_u \) is the degree of the vertex \( u \), and \( f(x,y) \) and \( g(x,y) \) are functions of the vertex degrees. Specifically, we apply \( X_\Sigma(G) \) and \( X_\Pi(G) \) on Erd\"os-R\'enyi graphs and random geometric graphs along the full transition from almost isolated vertices to mostly connected graphs. While we verify that \( P(X_\Sigma(G)) \) converges to a standard normal distribution, we show that \( P( X_\Pi(G)) \) converges to a log-normal distribution. In addition we also analyze Revan-degree-based indices and spectral indices (those defined from the eigenvalues and eigenvectors of the graph adjacency matrix). Indeed, for Revan-degree indices, we obtain results equivalent to those for standard degree-based TIs. Instead, for spectral indices, we report two distinct patterns: the distribution of indices defined only from eigenvalues approaches a normal distribution, while the distribution of those indices involving both eigenvalues and eigenvectors approaches a log-normal distribution.