The Punic Wars by Adrian Goldsworthy (dude knows more about the Roman Army than anyone)
Hannibal by Theodore Ayrault Dodge (Dodge was a Union officer in the Civil War and wrote some great books on Hannibal, Caesar, Alexander, etc. ... probably the best companion to primary source material on a first read through -- and it's out of copyright so you can find free copies online)
Cannae: The Experience of Battle in the Second Punic War by Gregory Daly (I haven't read this, but the slow trapping and butchery 70,000 men on a hot day seems like a fascinating topic for history as it was actually experienced)
Also, the History Channel has a pretty fun website, and there aren't any pawnshop aliens American Trucker-Pickers.
And if you want to read something that was written a tad earlier, Appian's histories cover the Second Punic War in several sections: The Spanish Wars, The Hannibalic War in Europe, and The Punic War and Numidian Affairs about Scipio in Africa (he also writes about the First Punic War), Livy deals with the Second Punic War in chapters 21-25 and 26-30, Polybius uses the Punic Wars to extol (and for us, explain) Roman virtues and institutions, and Plutarch gives two Generals treatment in his Parallel Lives, Fabius and Flaminius.
Oh... after you're done with all/any of that you might want to go buy Rome Total War and play as the Scipii. Extra points if you download Europa Barbarorum. Rome 2 is out and presumably awesome (and EB2)
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.
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:
Seconding the recommendation you got about the Peloponnesian Wars, and for the Punic Wars I would say try Adrian Goldsworthy's The Punic Wars. I haven't read that specific one but I've read some of his other works and they came recommended for pop-reading by a friend of mine who studies classics.
First off, remember that the civil wars in Rome had started a long time before Caesar's Civil War ;) But yep, you're correct! Probably the best book on the time period is Tom Holland's Rubicon. For a second suggestion, I'll add in Goldsworthy's (SHUSH) Caesar: Life of a Colossus. I'm drawing blanks on more books, but the books that are just as important as Rubicon, in my opinion, are Caius (Gaius) Julius Caesar's own works.
Teutoburg, I'm hesitent to advise too much on - I haven't done extensive research into it, myself. Sorry!
Adrian Goldsworthy wrote a book on the Punic Wars. He is an excellent writer and a great military historian. His book is quite comprehensive, however there is no other book that I would recommend to someone just starting out.
What are the best books on Hannibal (particularly ones that may have been overlooked)?
Personally I like Lazenby's Hannibal's War (for the academically inclined) and Dodge's Hannibal (for a general audience).
EDIT:
For those interested in learning more about Hannibal, here are my top picks from books actually on my book shelf:
And recommendations and from /u/gevemacd :
I scanned r/askhistorians for a minute and it seems that many people are recommending The Punic Wars by Adrian Goldsworthy.
Complete novice? Extra Credits.
Seconding the Podcasts from Carlin, "Punic Nightmares" and Duncan's History of Rome and Born Yesterday. Seriously, Duncan is amazing. Major history hard-on.
Also, the History Channel has a pretty fun website, and there aren't any pawnshop aliens American Trucker-Pickers.
And if you want to read something that was written a tad earlier, Appian's histories cover the Second Punic War in several sections: The Spanish Wars, The Hannibalic War in Europe, and The Punic War and Numidian Affairs about Scipio in Africa (he also writes about the First Punic War), Livy deals with the Second Punic War in chapters 21-25 and 26-30, Polybius uses the Punic Wars to extol (and for us, explain) Roman virtues and institutions, and Plutarch gives two Generals treatment in his Parallel Lives, Fabius and Flaminius.
Modern books, I liked Adrian Goldsworthy's [The Punic Wars] (http://www.amazon.com/The-Punic-Wars-Adrian-Goldsworthy/dp/0304352845), and had WAY too much fun reading this book about Scipio and this book about Hannibal in tandem.
Oh... after you're done with all/any of that you might want to go buy Rome Total War and play as the Scipii. Extra points if you download Europa Barbarorum. Rome 2 is out and presumably awesome (and EB2)
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.
Seconding the recommendation you got about the Peloponnesian Wars, and for the Punic Wars I would say try Adrian Goldsworthy's The Punic Wars. I haven't read that specific one but I've read some of his other works and they came recommended for pop-reading by a friend of mine who studies classics.
Hey, I'm glad to be of service! :D And again, if you need any more, please don't hesitate to ask. On to the points!
Adrian Goldsworthy wrote a book on the Punic Wars. He is an excellent writer and a great military historian. His book is quite comprehensive, however there is no other book that I would recommend to someone just starting out.
http://www.amazon.com/The-Punic-Wars-Adrian-Goldsworthy/dp/0304352845
Hope this helps!