You should check out the reading in the side bar, which can be found here: https://www.reddit.com/r/AskHistorians/wiki/books/europe#wiki_ancient_rome

Though honestly for a general overview of Rome from start to finish, Mary Beard's "SPQR" is fantastic. It's aimed at the general public rather than history students, which has its pros and cons. It's not free, but it's worth every penny.

If you're happy to spend a bit more money to get 3 books instead of one, all of which are aimed at new students of history, good introductions are Michael Crawford's "The Roman Republic", Collin Wells' "The Roman Empire", and Averil Cameron's "The Later Roman Empire". They are part of the same series that give an overview of the Roman Empire.

I have recommendations tailored for reference and for each period below.

Reference: Lewis and Reinhold's 2 volume series Roman Civilization, Selected Readings is an excellent source for all of Roman civilization. Their books are functionally a compilation of primary sources or early secondary sources and cover everything from the Roman Religion and its development to military conquests to roles of women in society. Another book that stretches across Rome's history and serves as an excellent guide is Mackay's Ancient Rome: A Military and Political Philosophy. It's a bit more cursory than Lewis and Reinhold, but it's also easier to read.

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For the Regal period and early republic, Livy's History of Rome is probably the best source. It's not primary, but he's the best extant source we have. I also recommend Plutarch's Lives for this purpose as well.

For the period of Rome rising to power over the ponent, Polybius is the premiere source. He's probably my favorite historian, though, so I'm biased.

For the tumultuous 1st century BCE, I recommend Sallust's Jurgurthine War and his record of the Cataline Conspiracy and Caesar's Commentaries on the Gallic War.

Two books I recommend reading to examine the course of the fledgling empire are Suetonius's The Twelve Caesars and Tacitus's Annals of Imperial Rome. Seutonius's work is more salacious and entertaining than Tacitus's, but his name is where the word tacit comes from after all.

For a look at the German Tribes of the time, Tacitus's Germania does the trick well.

For a peak into the rapidly changing lives of the nobility during the transition of late republic to early empire, I recommend reading M.P. Cato's Roman Farm Management: The treatises of Cato and Varro. You will definitely see contrast between the works.

For a look at medicine in the early empire, I recommend The Prince of Medicine by Susan Mattern. This is one of my favorites on the Roman Empire because you are able to visualize the life of people of various backgrounds around the Roman provinces, not to mention the life of the personal physician of the emperor and one of the founders of medicine.

For the Crisis of the third century (and eventual "fall"), I recommend Adrian Goldworthy's The Fall of The West: The Slow Death of the Roman Superpower.

For a look at the revival following the crises of the third century, I recommend reading Eusibius's Life of the Blessed Emperor Constantine.

For a closer inspection of religion towards the end of the Roman empire Julian's Misopogon and St. Augustine's City of God