Complexity and decidability of logical systems is a major research area currently involving a huge range of different systems from fragments of first-order logic to modal logic, description logics, et cetera. Due to the sheer number of different frameworks investigated, a systematic approach could be beneficial. We introduce a research program based on an algebraic approach to systematic complexity classifications of fragments of first-order logic and beyond. Our base system GRA, or general relation algebra, is equiexpressive with FO. The system GRA resembles cylindric algebra and Codd's relational algebra, but employs a finite signature with seven different operators. We provide a comprehensive classification of the decidability and complexity of the systems obtained by limiting the allowed sets of operators. Furthermore, we also investigate algebras with sets of operators not included in the list of the seven basic ones and use such operator sets to give algebraic characterizations to some of the best known decidable first-order fragments. To lift the related studies beyond FO, we also define a notion of a generalized relation operator. These operators can be seen to slightly generalize the notion of a generalized quantifier due to Mostowski and Lindstrom.