These books should cover the early period quite well, though you will have to wait for someone else to recommend some books for the later period! I should note at the outset that this field of study is hugely complicated and specialists often have very different views about the same topic. I will therefore try to provide some different perspectives here. To get a sample of the debates involved, Robert Hoyland has helpfully uploaded this article online. It only covers the difficulties concerning writing the biography of Muhammad, but the same problems exist basically for all aspects of the first century of Islam.

• Michael Cook, Muhammad (1983) - this is a bit dated, but this short introduction to what we know about the Prophet's life is still very useful. There are some pretty major issues within the field that hasn't been addressed by historians yet, so until a new critical biography is written, this short introductory book is probably the best we got.
• Robert Hoyland, In God's Path: The Arab Conquests and the Creation of an Islamic Empire (2014) - a very recent and very well-written book on the Arab conquests, provides a very up-to-date overview of the chronology and what modern historians think. Hugh Kennedy's The Great Arab Conquests: How the Spread of Islam Changed the World We Live in (2007) is useful as well, but I find that its narrative is a bit old-fashioned and not as 'cutting-edge' as Hoyland's book; both were written for a popular audience and so equally readable. Hoyland's book is also criticised by Fred Donner here. I'm not entirely convinced by this review, but it is worth bearing in mind when reading this essential book.
• Fred Donner, Muhammad and the Believers: At the Origins of Islam (2010) - presents an interesting, but not always convincing, narrative and explanation for the growth of early Islam. Includes however a narrative of both Muhammad's life and the conquests, so it is a useful way into this complicated field if Hoyland's book is unavailable.
• Hugh Kennedy, The Prophet and the Age of the Caliphates: The Islamic Near East from the 6th to the 11th Century (2004) - probably the best survey book for the period covered, great if you only want to read one book covering everything, but otherwise, I'd recommend starting from Hoyland then read a bit about the historiography.
• Fred Donner, Narratives of Islamic Origins (1998) - much more complicated than the other books I've mentioned, but it goes into some depths about why the sources for this period are so difficult to use. You might want to read this if you want to dig deeper into the historiography a bit more.
• Chase Robinson, Islamic Historiography (2003) - like the above book, read for an insight into the problems involved rather than for a narrative. Might be a better introduction than the above though.
• James Howard-Johnston, Witnesses to a World Crisis: Historians and Histories of the Middle East in the Seventh Century (2010) - this is a book for when you get really adventurous, as Howard-Johnston tries to examine all the sources for the seventh century here and attempts to constructive a narrative from the bottom-up. This is a phenomenal piece of scholarship, but his reconstructions are not always convincing and you have to go through a lot of very technical analysis to get to the narrative. Probably shouldn't be the first book you read on this period!

For some background on the world of late antiquity, I have two recommendations:

• Peter Sarris, Empires of Faith: The Fall of Rome to the Rise of Islam, 500-700 (2011) - does exactly what it says in the title, places the rise of Islam into its much larger context.
• Glen Bowersock, The Throne of Adulis: Red Sea Wars on the Eve of Islam (2013) - covers the wars in Arabia in the sixth century, which provides a very important background for the events of the next century. This is very readable as well, since Bowersock is a very engaging writer.

Shlin28 put out a good list, but I'd add a more recent book if you're up for it. I recently read The War of Three Gods, and it provided a good historical perspective on the rise of Islam, though it doesn't focus on religious details as much. It sums up the 7th century pretty perfectly and how the major events of the world influenced the Islamic Empire.