We develop a pipeline to mine complex drug interactions by combining different similarities and interaction types (molecular, structural, phenotypic, genomic etc). Our goal is to learn an optimal kernel from these heterogeneous similarities in a supervised manner. We formulate an extensible framework that can easily integrate new interaction types into a rich model. The core of our pipeline features a novel kernel-learning approach that tunes the weights of the heterogeneous similarities, and fuses them into a Similarity-based Kernel for Identifying Drug-Drug interactions and Discovery, or SKID^3. Experimental evaluation on the DrugBank database shows that SKID^3 effectively combines similarities generated from chemical reaction pathways (which generally improve precision) and molecular and structural fingerprints (which generally improve recall) into a single kernel that gets the best of both worlds, and consequently demonstrates the best performance.