Your question is really broad, so I'll attempt to give you a far-reaching answer concerning Achebe's Things Fall Apart.
There are numerous ways one could analyze this novel. I'll give you a list to choose from, based on the eNotes Study Guides on the novel.
- Custom and Tradition
- Choices and Consequences
- Alienation and Loneliness
- Change and Transformation
- Good and Evil
- Culture Clash
Characterization: methods and results
- Plot and Structure
- Foils: Okonkwo & Obierika, Ikemefuma & Nwoye, Mr. Brown & Reverand Smith
Again, there are endless ways to analyze any work of literature, but if you're looking for something concrete to begin your study with, the above should give you plenty of possibilities. I strongly suggest looking at the Study Guides I'll list below.
There are a variety of ways to approach this text; one of the most straight-forward is as a criticism of the European colonization of and exploitation of Africa and its peoples. The characters and story of Things Fall Apart are strong examples of the dangers of cultural engineering and the loss of innocence and even the loss of common sense that can come from the relentless push towards "progress" or someone else's ideal.
In combination with a book like King Leopold's Ghost, you can examine some of the historical assumptions and the commentary Achebe is making with the story. An in-depth knowledge of the time period and the various colonial powers and their actions along with the various cultures and nations within Africa can create a very powerful discussion of the way that Achebe represents the "white man" and the African alike.
The different approaches can be tailored to fit older students with more experience or understanding of the historical and cultural issues of the time or younger students new to these types of questions and without a great deal of background knowledge or understanding of the context.
Personally, i enjoyed analyzing the theme of nature. it is also interesting to study from a Biblical perspective