Social media are digitalising massive amounts of users' cognitions in terms of timelines and emotional content. Such Big Data opens unprecedented opportunities for investigating cognitive phenomena like perception, personality and information diffusion but requires suitable interpretable frameworks. Since social media data come from users' minds, worthy candidates for this challenge are cognitive networks, models of cognition giving structure to mental conceptual associations. This work outlines how cognitive network science can open new, quantitative ways for understanding cognition through online media, like: (i) reconstructing how users semantically and emotionally frame events with contextual knowledge unavailable to machine learning, (ii) investigating conceptual salience/prominence through knowledge structure in social discourse; (iii) studying users' personality traits like openness-to-experience, curiosity, and creativity through language in posts; (iv) bridging cognitive/emotional content and social dynamics via multilayer networks comparing the mindsets of influencers and followers. These advancements combine cognitive-, network- and computer science to understand cognitive mechanisms in both digital and real-world settings but come with limitations concerning representativeness, individual variability and data integration. Such aspects are discussed along the ethical implications of manipulating socio-cognitive data. In the future, reading cognitions through networks and social media can expose cognitive biases amplified by online platforms and relevantly inform policy making, education and markets about massive, complex cognitive trends.