Operator ideals on Hilbert spaces can be used to construct genuine infinite-dimensional geometric structures. In particular, Banach-Lie groups associated with operator ideals and their homogeneous spaces give rise to problems that are interesting for both operator theory and infinite-dimensional geometry. In this minicourse we will focus on some specific problems in the mentioned setting. We will consider the problem of diagonalization by unitary operators associated with operator ideals, propose a geometric study of the Moore-Penrose inverse and the polar decomposition of perturbations by operator ideals, and present aspects of abstract frame theory in Hilbert spaces related to operator ideals.