I have an earlier answer I wrote on book recommendations for more modern Central Asian history I'll copy a bit from, but to take a bigger perspective, here we go.

Rene Grousset's Empire of the Steppes is probably the foundational Western history on Central Asia through the 18th century. It's pretty old though (1939, first translated into English in 1952).

A more updated history that covers this period up to the post-Soviet era is Svat Soucek's History of Inner Asia. This is probably the most comprehensive single volume history covering the whole region that is out there and is a good introductory history.

Despite the reputation of most of the stuff it publishes, the Hoover Institution Press has a very good "Studies of Nationalities" series by experts in particular nationalities. Martha Brill Olcott's The Kazakhs and Edward Allsworth's The Modern Uzbeks are the most relevant here - Olcott in particular as her book is the most comprehensive history of Kazakhstan from ancient to post-Soviet times in English that I'm aware of.

S. Frederick Starr's Lost Enlightenment: Central Asia's Golden Age from the Arab Conquest to Tamerlane is a readable history of Medieval Central Asia. It's a better bet than Peter Frankopan's The Silk Roads, which despite its name doesn't really deal with Central Asia.

Adeeb Khalid is the go-to expert on the history of Islam in modern Central Asia, and he has written three books that cover that topic and are focused on Uzbekistan: The Politics of Muslim Cultural Reform: Jadidism in Central Asia covers the early 20th century, Making Uzbekistan: Nation, Empire, and Revolution in the Early USSR covers the early Soviet period, and Islam after Communism: Religion and Politics in Central Asia covers the late Soviet period and history through independence to roughly 2000 or so.

For the Russian colonial period, there is no shortage of histories. Daniel Brower edited Russia's Orient: Imperial Borderlands and Peoples, 1700-1917, Michael Khodarkovsky's Russia's Steppe Frontier (this covers the Central Asian steppes, while William Sunderland's Taming the Wild Field covers similar history in the European steppes), and Ian Campbell's Knowledge and the Ends of Empire: Kazak Intermediaries and Russian Rule on the Steppe, 1731-1917 being a very specific entry for the experience in Kazakhstan.

For more specific Soviet era history: Kazakhstan has a few new histories out, such as Sarah Cameron's The Hungry Steppe and Robert Kindler's Stalin's Nomads: Power and Famine in Kazakhstan, both of which cover the 1930s famine. Mukhamet Shayakhmetov's The Silent Steppe is a Kazakh memoir of this era. Adrienne Edgar's Tribal Nation: The Making of Modern Turkmenistan covers a similar period for Turkmen, and Douglas Northrop's Veiled Empire: Gender and Power in Stalinist Central Asia covers the Soviet anti-veiling campaign in Uzbekistan during that same period. Roberto Carmack's Kazakhstan in World War II: Mobilization and Ethnicity in the Soviet Empire is the only specific World War II history of the region that I'm aware of, and it's very new.

For post-Soviet history:

Olivier Roy's The New Central Asia: Geopolitics and the Creation of Nations is worth a read. Roy was a French diplomat in Tajikistan in the 1990s, and this book is largely primarily on Tajikistan (but Uzbekistan plays a major role) and also more on the sociology of the region than just a straight-up political history. Feel free to skip his last chapter on conclusions since they're woefully out of date.

You also might want to check out Alexander Cooley and John Heathershaw's Dictators Without Borders: Power and Money in Central Asia has a chapter on each Central Asian country, highlighting a major international corruption scandal in each. Strictly speaking they are political scientists and not historians, and are not covering history of the countries per se, but this book does a good job of situating what high-level corruption in the region looked like post 1991, and how international institutions both aided it and were abused by it.

Some books on post 1991 history that I'm less familiar with: Pamela Blackmon's In the Shadow of Russia: Reform in Kazakhstan and Uzbekistan, Dilip Hiro's Inside Central Asia: A Political and Cultural History of Uzbekistan, Turkmenistan, Kazakhstan, Kyrgyzstan, Tajikistan, Turkey, and Iran (note: this is probably a good introductory modern history, but a lot of Hiro's translations were so bad and irritating I gave up reading it - a reader not familiar with Central Asian languages probably won't have this problem), and Pauline Jones Luong's (as editor): The Transformation of Central Asia : States and Societies From Soviet Rule to Independence.

Anyway, that's all just some books to get started. Also if there are specific history follow up questions, let me know, even something like "pre-Mongol Kazakhstan" is a vastly wide topic and period.