This is a very broad question. There is no one way societies interact. There are many variables. In fact, there are too many variables that general statement are hard to make. That said, here are some common ways in which societies interact. Let me mention just two of them, which you can flesh out yourself.
First, there is always the possibility of war and conflict. As long as nations, tribes, or societies existed, there has been conflict. Wars have always marked humanity. Even when we are at peace, there is war in subtle ways - influence, worldview, and various ideologies.
Second, groups and even nations can work together to form alliances. In a complex world as ours, alliances are a common way in which societies interact. Usually those in the alliance all benefit in some way.
Another way societies can interact is through disease exchange. During the 1300s, societies across the Silk Roads interacted through the exchange of the Black Death. Another example would be the interaction of societies after WWI through the influenza epidemic.
Societies interacts in various ways such as in trade/barter/sale relationships, or in forming alliances in times of conflict or war. Societies may also interact when religious fervour expands from one society's homeland to another society's homeland or continent such as occurred when the European States encroached on the Indigenous peoples of the Americas in the 1500's. Interacting societies may each benefit from one another such as in varieties of food production, inventions, communication tools (literature, books, dance, music), and in other mutual interactions such as inter-marriage, commerce, and constructive alliances.