This book looks at the history of the United States as told by people who came to the country in search for a better life. It is not strictly about immigrating to the US in itself, but rather why and how they came here and what challenges, successes, prejudices, etc. they encountered while trying to make their way on this land.
It is quite dense (in a good way) in the text of the groups that came (or how they adapted/coped in the case of Native Americans). There is a great deal of ground to cover and obviously it's not possible to do all of them justice. But there is probably a great deal to learn. I knew about the Chinese men who came and their struggles with being unable to bring their wives (in contrast to many of the Chinese women who were brought over and subjected to sexual slavery) but I did not know many of the stories like the Japanese in Hawaii as one example.
In many ways the book is great in giving us snapshots of various groups, providing historical context that is likely just not taught unless you take a special class or happen to read about it in a book. But as others point out sometimes it can get formulaic (group moves to the United States, is the target of discrimination and struggles/adapts). Which is rather horrible in itself repeating over and over again. The framing device in using Shakespeare's 'The Tempest' and Caliban in particular is quite annoying (since I haven't read this play or watched any adaptation of it I kept wondering why Takaki kept mentioning Caliban/'The Tempest').
But if you liked Howard Zinn's 'A People's History Of The United States' you might like this. It's actually been many MANY years since I've read Zinn's book but as I read I couldn't help but think of that text. Takaki's book would make a good compliment but as mentioned it's a rather thick book.
Some other recommended readings to go along with this (perhaps to expand) would include: 'The Warmth of Other Suns: The Epic Story of America's Great Migration', 'The New Jim Crow: Mass Incarceration in the Age of Colorblindness', 'The Making of Asian America', and 'Everything You Wanted to Know About Indians But Were Afraid to Ask'. There are probably tons more really great books but those are some I can think of off the cuff. That said, this is a book that you can probably read on its own but you might get more out of it if you have other sources to compliment your reading.