Reddit mentions of A History of the Roman Republic

The standard textbook history right now appears to be The Romans: From Village to Empire.

Klaus Bringmann's A History of the Roman Republic also still seems to be the standard introduction to that period (i.e. the time period of Imperator).

If you want to read about the end of the Roman Republic and Caesar/Augustus, it's hard to turn down Caesar: Life of a Colossus which is great for the general reader, alongside his Augustus: First Emperor of Rome.

He also writes pretty solid books on other major Roman figures, such as In the Name of Rome: The Men who won the Roman Empire.

If you want to get a pretty good introduction to Roman History, but more of what life was like for the average citizen, SPQR by Mary Beard is actually a good choice.

Older, but still solid, is Peter Garnsey's The Roman Empire: Economy, Society, and Culture which covers a lot of things Beard doesn't.

For the Roman army, Adrian Goldsworthy's The Complete Roman Army is a solid introduction.

However you'll want to break that down into several books if you want to go deeper:

Roman Military Equipment by MC Bishop and JCN Coulston

The Roman Imperial Army of the First and Second Centuries AD by Graham Webster

A Companion to the Roman Army by Paul Erdkamp

For the collapse of the Western Roman Empire I'd recommend both Peter Heather's The Fall of the Roman Empire: A New History of Rome and the Barbarians combined with the more scholarly Guy Halsall's Barbarian Migrations and the Roman West.

For the forgotten half of Roman History, often mistakenly called the "Byzantine Empire," it's hard to cover with just one book, but Warren Treadgold's A History of the Byzantine State and Society has become the standard reading. John Haldon's The Empire that would not Die covers the critical transition during the Islamic conquests thoroughly.

Of course I have to include books on the two IMO most overrated battles in Roman history on this list since that's what people love:

The Battle of the Teutoberg Wald: Rome's Greatest Defeat by Adrian Murdoch

The Battle of Cannae: Cannae: Hannibal's Greatest Victory is sort of the single book to read if you can only pick one. However, The Ghosts of Cannae is also good. But if you actually want to go really in depth, you need Gregory Daly's dry-as-the-Atacama book Cannae: The Experience of Battle in the Second Punic War. When I say dry as the Atacama, I mean it, but it's also extraordinarily detailed.

I'd complement this with Goldsworthy's The Punic Wars.

For other interesting topics:

The Emergence of the Bubonic Plague: Justinian's Flea and Plague and the End of Antiquity.

Hadrian's Wall: Hadrian's Wall by Adrian Goldsworthy

Roman Architecture: Roman Architecture by Frank Sear (definitely a bit more scholarly but you can probably handle it)

I may post more in addendum to this list with further comments but I think I'm reaching the character count.

I recommend Klaus Bringmann's A History of the Roman Republic. I also recommend a reader, such as Dillon and Garland's Ancient Rome: From the Early Republic to the Assassination of Julius Caesar. These two books together should give you an excellent look at the Republican era.

Ronald Syme's Roman Revolution would be my 'scholarly 'recommendation.

Chronicle of the Roman Republic/Empire would be my recommendation for a true introductory look and for use as a quick reference when reading more in depth texts, plus its pretty.

A good middle of the road intro text to the republic although the translation is a little clunky

This book also comes highly recommended but I haven't read it myself.