Generating realistic human motions that naturally respond to both spoken language and physical objects is crucial for interactive digital experiences. Current methods, however, address speech-driven gestures or object interactions independently, limiting real-world applicability due to a lack of integrated, comprehensive datasets. To overcome this, we introduce InteracTalker, a novel framework that seamlessly integrates prompt-based object-aware interactions with co-speech gesture generation. We achieve this by employing a multi-stage training process to learn a unified motion, speech, and prompt embedding space. To support this, we curate a rich human-object interaction dataset, formed by augmenting an existing text-to-motion dataset with detailed object interaction annotations. Our framework utilizes a Generalized Motion Adaptation Module that enables independent training, adapting to the corresponding motion condition, which is then dynamically combined during inference. To address the imbalance between heterogeneous conditioning signals, we propose an adaptive fusion strategy, which dynamically reweights the conditioning signals during diffusion sampling. InteracTalker successfully unifies these previously separate tasks, outperforming prior methods in both co-speech gesture generation and object-interaction synthesis, outperforming gesture-focused diffusion methods, yielding highly realistic, object-aware full-body motions with enhanced realism, flexibility, and control.