Accurate simulation of the low-energy physics of strongly correlated quantum systems is increasingly more within reach. The study of collective behaviour in these systems is one of the main focal points within condensed matter physics, due to its inherently fascinating nature, wide potential for applications and notorious difficulty. In this thesis, we present a combination of new results on the ground states and low-lying excited states in various models and the methodological advances that we needed in order to obtain them. We base our methods on the variational class of infinite projected entangled-pair states (PEPS), a type of two-dimensional tensor network that can represent quantum states directly in the thermodynamic limit. With PEPS, we investigate the ground-state phase diagram of the well-known Hubbard model in the presence of a next-nearest neighbour (NNN) hopping, and find a competition of stripe states with varying periods, in agreement with numerical and experimental data. Beyond ground states, we present new implementations of the PEPS excitation ansatz, a powerful framework for simulating low-lying excited states as small perturbations to the ground state, providing results on spin models and a free fermionic model that agree well with existing data. Furthermore, we show how automatic differentiation can be employed to create an even more generally applicable and efficient variant, which we apply to the charge excitations of the half-filled Hubbard model. The results compare well to new and existing quantum Monte Carlo data, further establishing our method as a competitive and reliable tool for future studies.