In this dissertation a newly developed numerical method is presented, which is optimized for the time evolution of one-dimensional lattice systems with large local Hilbert spaces. This method extends the time-evolving block-decimation (TEBD) to include a local basis optimization (LBO), which has already been successfully combined with ground state methods. The algorithm is based on matrix product states (MPS), which can represent the quantum state of a one-dimensional chain in most cases with a number of parameters, that is not exponentially increasing with the chain length. The LBO causes a reduction of the simulation times that is linear in the bond dimension of the MPS. To demonstrate the advantages of this method, we apply the TEBD-LBO to electron-phonon (e-p) systems. In this thesis these are described by the Holstein model, which goes beyond semi-classical approximations and covers the full quantum statistics of the phonons. The understanding of the nonequilibrium dynamics of charge carriers coupled to lattice vibrations is of great importance for research areas like transport through quasi one-dimensional conductors, photo-generated phase transitions and time-resolved spectroscopy. First, the energy transfer from a highly excited electron to the phononic degrees of freedom on a small chain is studied. In the various parameter regimes different types of relaxation occur. In any case, after a certain time the system reaches a state, where on average no energy is exchanged between the electron and phonons. This can either mean a constant kinetic energy or oscillations with a constant amplitude and frequency. Next, long perfectly conducting leads without coupling to phonons were attached on both sides of the small chain. In the left lead an electron with density distribution in the shape of a Gaussian wave packet is injected with momentum towards the e-p coupled structure in the middle. This structure acts as an impurity in an otherwise perfectly conducting chain. The investigation shows resonance effects in the transmission and reflection at this impurity. Further, the electron can transfer a part of its energy permanently to the phonons, which results in a reduction of the velocity. Finally, two mechanisms are presented that lead to self-trapping of the electron on the e-p coupled structure.