For the modern person perplexed by the globalized culture around him:

The Kite Runner by Khaled Hosseini - depicts the tumultuous history of contemporary Afghanistan and touches on a lot of different social issues like race, immigration, war, religion, women in Muslim cultures and some very important personal issues like guilt and fear.

The Time Machine by H.G. Wells - Another short tale that is a Science Fiction legend, and a great fictionalization about dense scientific concepts like time. Not only that, but the narrative implicitly explores themes of colonization and socioeconomics, which other British authors in the 19th century were doing. This book is highly accessible.

Postmodernity, or, the Cultural Logic of Late Capitalism by Frederic Jameson - Very dense and broad, and less of a book and more of an article, this piece will help you unpack how culture has changed since the 1980s and posits some good theories regarding some of the functions of our culture (weakening of historicity, death of meta narratives, "schizophrenia" in art)

Mrs. Dalloway by Virginia Woolf - Not everyone likes this one, but it's a good read for the sake of accurately depicting the disillusionment with the world in modernity following post WW1. Woolf killed herself in the 40s - and the titular character of this book really expresses some of her frustrations with the world without being preachy.

Maus by Art Spiegelman - Out of the wide variety of books you can choose regarding the Holocaust, I love this one for its creativity (it's an autobiographical comic) and the characters self-reflexivity. A mix between Man vs Author and Man vs Society.

I could go on and on including more philosophy/theory, drama, poems, etc. But these are pretty good starts and focus on a lot of the issues that are raised in contemporary fiction anyway.