Reading the books in addition to watching the show is a great way. The books often have more in - depth backstories and details. For instance, the history of the Iron Islands and their culture is more fully explored in the books, which can reveal hidden stories related to the Greyjoys. Also, analyzing the symbolism in the show, like the significance of the direwolves, can lead to uncovering hidden stories.
Look at the minor characters. Sometimes they hold the keys to hidden stories. Take Gendry for example. His parentage and his role in the story had hidden aspects that were slowly revealed. Another way is to study the maps and the different regions. The politics and history of each region, like Dorne, has hidden stories that can be uncovered by understanding their geography, culture and past events.
You can start by paying close attention to the dialogue. There are often hints and foreshadowing in what the characters say. For example, Littlefinger's cryptic remarks sometimes pointed to hidden plots.