Archaeologists call the time just after people first came to North America the Paleo-Indian period. During the Paleo-Indian period, all the people in North America were nomads and got their food by hunting and gathering. Some of these people stayed in Alaska.
Other people gradually went south and east, looking for warmer weather, little by little. They hunted the mammoth as they went, and then as the mammoth became extinct they began to hunt buffalo instead. They used spears and atlatls to hunt. After a while (nobody is sure how long), some people reached Central America and then South America, while other people reached the east coast of North America.