"I, Claudius" has rapidly become my second favorite historical fiction. It's written by the author as an 'autobiography' of the Emperor Claudius, who had all sorts of troubles in his life -- he was lame and could barely walk, was bullied often as a child and overlooked, his father was most likely killed by poisoning at a young age, he stuttered, and had a variety of other issues. And yet, he survived and became Emperor.
I want to recommend it, since it's excellent especially on audiobook (the narrator is awesome, hilarious, does voices well, understands drama, and obviously knew Roman history well enough himself to cover it very credibly).
But the more I thought about recommending the book, the more I realized you need a lot of background on Roman history to truly appreciate it. It's a great book for weaving together the pieces of Roman history from the end of Republic through to the establishment of the Empire, but you need the background on the Republic, Civil Wars, and early Empire first.
So I thought about it. Here's my recommended order for learning some Roman history, with a mix of links to podcasts, books, and audiobooks --
1. Hardcore History's The Death of the Roman Republic series: Hardcore History is my favorite podcast, with Dan Carlin really bringing history to life. This is the best place to dive into Rome, adn it explains all the tensions and conflict of the late Roman Republic which led to the Civil Wars, introduces you intimately to many of the personalities involved, and is really enjoyable and exciting in the process. It's entirely free, so start here.
2. Learn about Augustus: Roman history (and Claudius in particular) doesn't make sense until you understand about what Augustus did, and that's where Hardcore History ends the series. I can grudgingly recommend Anthony Everitt's "Augustus: The Life of Rome's First Emperor," which is adequate but not great. While the writing and scholarship is good, it just doesn't captivate the way some of the other titles on this list do... but Augustus's life is so interesting and important that maybe the writing stands up anyways. You could alternatively just read some summaries and essays on Augustus's life. After following from your understand of why the Roman Civil Wars happened (Hardcore History will explain it well), Augustus's life and reforms are the next logical and important place to go. Augustus was the architect of the Roman Empire, and had just a tremendously huge effect on the next 500+ years of Roman history.
3. Read B.H. Liddel-Hart's Scipio Africanus: Well, now you're in for a treat. This short, fast biography is a masterpiece. Scipio Africanus defeated Hannibal and subdued Carthage, which paved the way for undisputed Roman dominance of the region. The biography is a masterpiece, and might be the best thing on this list. You'll understand a lot of Roman military tradition and hegemony after reading this. These events come around 200 years before the late Republic, but is crucial to understand when all the Romans are lamenting the fall of Roman virtue, the corrupting of the offices, etc.
4. Listen to Julius Caesar's autobiography: Caesar wrote an autobiographical accounts of his wars in Gaul and Britain, and it's a masterpiece of clear writing and thinking. It gets a little boring in text since it's repetitive: Caesar is detailing how he prepares for battles and executes them, which is often similar -- secure food, send out envoys and diplomats, scout the area, do military engineering, dig in for battle, outlast the enemy. Yet, it's great in training you how to think. I strongly recommend the audiobook version of it, since one of my favorite narrators does it, and it really comes alive in voice as opposed to text on the page.
5. Brush up on Roman offices and military a bit: Between Hardcore History, Augustus, Scipio, and Caesar, you should know the big pieces. I'd do a Wiki-walk around Wikipedia about Rome now, starting with "Roman Republic" (linked above), and following interesting people, the political offices, the various wars, personalities, etc.
6. I, Claudius: You could even start with this, and that'd be fine. But you won't understand the significance of a lot of events. Why did Augustus's wife want such-and-such's grandson dead? Why was Augustus able to survive as a de facto monarch whereas Julius Caesar was assassinated? The book is much richer if you understand and are grounded in Roman history a bit first. It's a good book without them, but if you follow the first five steps (all of tremendously enjoyable in and of themselves), then the text really comes alive and will weave together your understand of Roman history.
That should cover your first 70 hours of learning about Roman history.
Never the less, I'd start with the free Hardcore History podcast which is excellent, brush up on Augustus briefly, read Scipio Africanus, and then get into audiobook for Caesar's Commentaries and I, Claudius -- both of which are excellent.