The study of quantum chromodynamics (QCD) at finite temperature and density provides important contributions to the understanding of strong-interaction matter as it is present e.g. in nuclear matter and in neutron stars or as produced in heavy-ion collision experiments. Lattice QCD is a non-perturbative approach, where equations of motion for quarks and gluons are discretized on a finite space-time lattice. The method successfully describes the behavior of QCD in the vacuum and at finite temperature, however it cannot be applied to finite baryon density due to the fermion sign problem. Various QCD-like theories, that offer to draw conclusions about QCD, allow simulations also at finite densities. In this work we investigate two-color QCD as a popular example of a QCD-like theory free from the sign problem with methods from lattice gauge theory. For the generation of gauge configurations with two dynamical quark flavors in the staggered formalism with the rooting trick we apply the Rational Hybrid Monte Carlo (RHMC) algorithm. We carry out essential preparatory work for future simulations at finite density. As a start, we concentrate on the calculation of the effective potential for the Polyakov loop, which is an order parameter for the confinement-deconfinement transition, in dependence of the temperature and quark mass. It serves as an important input for effective models of QCD. We obtain the effective potential via the histogram method from local distributions of the Polyakov loop. To study the influence of dynamical quarks on gluonic observables, the simulations are performed with large quark masses and are compared to calculations in the pure gauge theory. In the second part of the thesis we examine aspects of the chiral phase transition along the temperature axis. The symmetry group of chiral symmetry in two-color QCD is enlarged to SU(2 Nf). Discretized two-color QCD in the staggered formalism exhibits a chiral symmetry breaking pattern of U(2 Nf)->O(2 Nf), contrary to the continuum theory. We determine pseudo-critical couplings where Ferrenberg-Swendsen reweighting is applied for an improved extraction of the peak of the chiral susceptibility. In order to assess the universality class critical exponents are studied via the scaling behavior of the chiral condensate and the corresponding susceptibility. Simulations are performed at various small quark masses to obtain results in the chiral limit. By introducing an improved discretization of the gauge action we mitigate effects of an unphysical phase, which appears as a discretization artifact at small values of the lattice coupling. Furthermore, an important step is the detailed investigation of finite volume effects, which become relevant at very small quark masses. When temperature is varied using the coupling constant, also the underlying length and energy scale is modified. It is desirable to simulate along lines of constant physics (LCP) in parameter space. We thus have begun to calculate meson masses to determine LCP via the pion to rho meson mass ratio. Influence of the bulk phase at low lattice couplings and finite-volume effects at larger couplings however hamper their calculation.

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