How do I actually explore a dataset instead of just running stats?

Summary stats tell you the averages; exploration tells you the story, and it is normal to have one before the other. Averages hide as much as they reveal. Real exploration means getting curious about what is underneath them. Start asking questions of the data instead of just describing it. How do groups differ? What changes over time? Where are the extremes, and why? Draw simple pictures, because your eye catches patterns that a table of numbers hides. Follow whatever looks strange; that is usually where something is. One small step: take one dataset and make three quick charts asking three different questions. Looking, not just calculating, is the habit that turns numbers into meaning.