Graphical event models (GEMs) are representations of temporal point process dynamics between different event types. Many real-world applications however involve limited event stream data, making it challenging to learn GEMs from data alone. In this paper, we introduce approaches that can work together in a score-based learning paradigm, to augment data with potentially different types of background knowledge. We propose novel scores for learning an important parametric class of GEMs; in particular, we propose a Bayesian score for leveraging prior information as well as a more practical simplification that involves fewer parameters, analogous to Bayesian networks. We also introduce a framework for incorporating easily assessed qualitative background knowledge from domain experts, in the form of statements such as `event X depends on event Y' or `event Y makes event X more likely'. The proposed framework has Bayesian interpretations and can be deployed by any score-based learner. Through an extensive empirical investigation, we demonstrate the practical benefits of background knowledge augmentation while learning GEMs for applications in the low-data regime.