We establish limit theorems that describe the asymptotic local and global geometric behaviour of random enriched trees considered up to symmetry. We apply these general results to random unlabelled weighted rooted graphs and uniform random unlabelled $k$-trees that are rooted at a $k$-clique of distinguishable vertices. For both models we establish a Gromov--Hausdorff scaling limit, a Benjamini--Schramm limit, and a local weak limit that describes the asymptotic shape near the fixed root.