Keep in mind that, in order to understand this work, the reader has to have a grasp of its organization. The major parts of the work are the parts you mention. The work is further delineated into twenty-four numbered sections. Each section is then divided into three separate stories: one mythological, one historical, and one personal. Your question, then, involves only the second part: the historical.
The "experiences" you ask about have to do with the Kiowa tribe and sometimes have very specific dates attached to them. The first major part, "The Setting Out," is, as the name suggests, about the creation of the Kiowa tribe. According to the different historical stories (which make up the middle of each of the numbered sections), "The Setting Out" involves Kiowa prehistory; therefore, the specific date of creation cannot be given. However, it is probably sometime in the early 1600s, before the European settlers would have ventured near the Great Plains. Of course, it could possibly be even earlier.
The second major part, "The Going On," is about the inner workings of the Kiowa tribe and has to do with the tribe's heyday: from its creation to the early 1800s. The Kiowa have a beautiful relationship with the horse, which allows them to migrate from Montana all the way to the Mexican border, where they see "men with tails," which supposedly are monkeys. In this section, the Kiowa escape from many enemies during their migration.
The last major part of the book is called "The Closing In" for good reason: it represents the metaphorical "end" of the Kiowa tribe. This is the major part that has most of the dates embedded within it. Specifically, it involves the 1800s and, particularly, the late 1800s. During the nineteenth century, the Kiowa were pummeled by war, disaster, and division. The buffalo were killed mercilessly. The horses were massacred. A meteor shower proclaimed it all. In 1875, the American troops declared an end to fighting with Native Americans on the Great Plains when the Kiowa slowly surrendered at Fort Sill. It is soon after this, in 1887, that Momaday's grandmother (Aho) performs in the very last Sun Dance of the Kiowa. The nail in the Kiowa coffin came in 1890 when American troops ordered the Kiowa to disperse. This ended any possible gathering of the Kiowa tribe.
In conclusion, even though you are asking about specific "time periods" here, keep in mind that this book is really about language more than time periods. The Kiowa language is what truly connects the major parts (and the smaller numbered sections) of this book. In other words, it is language that is of primary importance, even more than historical dates.