Recommendation for an introductory history of ancient Rome?

Okee doke, I have two things for you! First off: I just made a post with recommended books over here yesterday :) I'll quote it for you below!

Starting off, my favourite authour with regards to Roman history is Adrian Goldsworthy. His books aren't only readable - they're extraordinarily informative and entertaining, all rolled into one. He's written LOADS of books on Rome, and every single one has been a treat - I can't recommend him highly enough.

Regarding the Roman military, there are a couple of books that he's written that'll outline the Roman military WONDERFULLY for you - hell, they definitely were highly useful for me. The Complete Roman Army is exactly what it says - it's a fantastic book that's not too long, but is completely chock full of information and pictures (It sounds weird that I emphasize this, but seriously. Those pictures are AWESOME). It primarily focuses on the force that we think of when we think "Roman Army" - the army of the Principate - but it outlines the Roman army through time.

Next, Roman Warfare. It's similar to the previous book, but focuses on the Roman army in a slightly different way, if that makes sense :) I highly recommend both books if possible.

Finally, with regards to the Roman army and how it helped grow the Republic/Empire, I'll recommend In the Name of Rome. This one focuses on the generals for the most part, giving short biographies of each, as well as a fantastic background of Rome herself, and how that evolution both influenced and was influenced by these men.

Now, to the evolution of Rome herself! Since you're familiar with the early republic (sorta), I'd advise reading The Fall of Carthage - it encompasses a lot more than the name suggests, as Dr. Goldsworthy LOVES to describe the background to help you understand the subject intimately - something few people do as well as he does. He begins with Alexander's death in ca. 323 BCE and continues through the legacy of the destruction of Carthage (which occurred in 146 BCE). Want to understand Rome between that time frame? Read this book.

Then, what I consider to be Goldsworthy's finest work - his biography, Caesar: Life of a Colossus. As the name suggests, the book discusses Caesar's life - and, as per Goldsworthy, it discusses everything about the society of his life. It's an incredibly good read and I consider it to by my "bible" of sorts with regards to the late Roman Republic. If you can only get one book on this list, this is the book that I recommend.

I haven't read anything regarding the Empire (ca. 32 BCE-476 CE), sadly, so I'll leave it there - however, you may be interested in one of Goldsworthy's books that I haven't read quite yet - How Rome Fell: Death of a Superpower

I hope that helps! :)

EDIT: One more! Barry Strauss' The Spartacus War is a rather brilliant read that gives some great insight into slavery, gladiators, and the Third Servile War (~72 BCE).

Next, I noticed that another user mentioned Dan Carlin. I'm gonna go ahead and straight up say that Dan Carlin is not a good way to introduce yourself to Ancient Rome. I'm sorry, but his massive, 5 part podcast is glorified pop history - while it's right in some ways, in others, it's completely misinformed, the main source he uses is Tom Holland's Rubicon (Which is essentially a modern day political commentary couched in pop history), and one of the worst things you can do is to start off by learning misinformation. Mr. Carlin, as he often says, is not a historian - and I actually did a writeup of my thoughts on him here :)

Hope that helps! If you have any more questions, I'll be happy to answer them :)