We report a detailed theoretical study of the electronic structure, spectral properties, and lattice parameters of bulk FeSe under pressure using a fully charge self-consistent implementation of the density functional theory plus dynamical mean-field theory method (DFT+DMFT). In particular, we perform a structural optimization and compute the evolution of the lattice parameters (volume, c/a ratio, and the internal z position of Se) and the electronic structure of the tetragonal (space group P4/nmm) unit cell of paramagnetic FeSe. Our results for the lattice parameters obtained by structural optimization using DFT+DMFT are in good quantitative agreement with experiment, implying a crucial importance of electron correlations in determining the correct lattice properties of FeSe. Most importantly, upon compression to 10 GPa our results reveal a topological change in the Fermi surface (Lifshitz transition) which is accompanied by a two- to three-dimensional crossover and a small reduction of the quasiparticle mass renormalization compared to ambient pressure. The behavior of the momentum-resolved magnetic susceptibility χ(q) shows no topological changes of magnetic correlations under pressure but demonstrates a reduction of the degree of the in-plane (π,π) stripe-type nesting. Our results for the electronic structure and lattice parameters of FeSe are in good qualitative agreement with recent experiments on its isoelectronic counterpart FeSe1-xSx.