This thesis aims at furthering effective Feynman-diagramatic methods for describing correlation effects of electrons in solids. Exact solutions for interacting electron systems with more than a hand full of particles are far beyond the reach of numerical methods. Therefore, approximations are needed. In this regard, effective Feynman-diagrammatic methods, such as the dynamical vertex approximation, use simple auxilliary models, which can be solved numerically. From these, diagramatic quantities are extracted which are used as approximations for their respective counterparts for the system of interest. Hitherto, such methods were for the most part based on effective oneand two-particle diagrams. This thesis expands upon this by providing an expression for n-particle vertices (correlators) for arbitrary numbers of particles n in the Falicov-Kimball model. Three-particle diagrams for the Falicov-Kimball and Hubbard models are evaluated exploratively, showing relevant corrections to the conventional two-particle calculations. Additionally, a systematic topological classification of three-particle diagrams is pursued, culiminating in an algorithm to calculate the most fundamental three-particle diagram: the fully irreducible three-particle vertex. Finally, the effect of the outer self-consistency for the Hubbard model within the dual fermion framework is investigated. This allows for understanding the influence of the auxilliary model better and shows how to update it iteratively. Maybe not surprisingly, the choice of auxilliary system has a strong influence on the theoretical description of correlated electron systems.