Study the history of ancient Western civilization by looking at the art that was made over the course of the centuries. Whether you look at Byzantine Jewelry, Greek statues or Roman architecture, the art of a culture can be a wedge into its history. The medium of art (metal or clay, for instance) as well the content of the art (geometric patterns or political figures, for instance) says a lot about the the resources and technologies available to the culture, as well as its values and politics.
The philosophy of ancient Western civilizations has been revered ever since its inception. It is, after all, one of the major foundations for Western thought today. It can also reveal a lot about the ways of the times. Learning about Aristotle, Socrates, Pythagoras and others can teach us about ancient ideas of science, math, reason and politics. However, things get more interesting when we question relationships between past and present ways of thinking: Is Western patriarchy today a result of male dominance in ancient philosophy? How do past and present medical practices compare?
Political history often focuses on wars, political turmoil, state relations within a region and important leaders. Learning the names and dates of various turning points in ancient Western civilized history helps a lot with developing a solid background for other research. You might zoom in on the transition from city-states to empire, or the trade relationships between Western civilizations and their neighbors. Try to focus on the areas you find the most interesting, and do a basic survey study of the parts in between.
Shift the focus of study away from the big names and political motivation of the state and shine it instead on the everyday people of the civilization. There is much to be learned about the political movements of the people, their lifestyle habits, their social relationships and their attitudes. Using these ancient peoples as examples, we can start to draw conclusions about our own civilization and its relative positives and negatives. Finding literature and primary evidence that helps you piece together people's history instead of relying on tales of the leaders can reveal a very different historical narrative.