I wanna learn more about Rome — Ancient Rome in the beginning, the Roman Republic, the Empire, the Fall, and all that stuff. What’s out there, what’s pretty readable?
The Immense Majesty. Dense and comprehensive, but enjoyable.
Rubicon: Last Years of the Roman Republic. More readable, but narrowly focused.
Well the big fat gold standard is of course The History of The Decline and Fall of the Roman Empire by Edward Gibbon. It’s got 6 (VI) volumes.
And of course there’s I, Claudius (which is fiction, keep in mind) or anything else by Robert Graves.
You might want to look into Jacques Barzun. What I’ve read of him is mostly post-Rome, but boy does he write fat (and really good) history book.
If you’re interested in fiction, Colleen McCullough’s (sp? The Thorn Birds author) Rome series is good.
For the later period you could do far worse than Pagan And Christian Rome by Rodolpho Lanciani. It’s a fascinating survey of the late Empire, with particular emphasis on the early Christian Church and its relations with the surrounding community.
If you have access to a large academic library you can probably find a copy there, or you can read the whole thing online here.
Try “The Provinces of the Roman Empire” by Theodor Mommsen. I got a hernia carrying it to the computer. But it may be antiquated. It was first published in 1885.
But as we know, Ancient Rome has changed tremendously since then. 
Actually, that’s an interesting question. How much has our evaluation of Rome changed based on our own historical perspective? I seem to remember reading that Gibbon, looking back from the 18th century, said Rome fell because it got soft directly or indirectly as a result of Christianity and didn’t have the moral turpitude and ruthlessness to do whatever it was that Romans did. Now, of course, we think Gibbon was a gasbag — thorough, perhaps, and dedicated to using primary sources, but off the mark on his reasoning, yes?
I stopped by to recommend McCullough, but I see Katriona got here first. It’s an excellent series, some of the best historical fiction I’ve read. IIRC, McC is a classics prof at a university in Australia and she knows her onions when it comes to ancient Roman history and culture.
If you want a good perspective on what we now understand as the history of Rome, I would not recommend Gibbon or Mommsen - they’re important for scholarship, but very outdated.
Here are some of my favourite overview-type books (non-fiction):
T. J. Cornell - The Beginnings of Rome (Pre-Roman Italy to the Punic Wars)
Erich Gruen - The Last Generation of the Roman Republic (Sulla to Caesar)
Sir Ronald Syme - The Roman Revolution (Pompey to Augustus)
Peter Brown - The Rise of Western Christendom (Late Antiquity)
There are a couple of chronological gaps in there (Punic Wars to Sulla; the vast majority of the Empire) but I can’t think of any good overviews at the moment. Brown I didn’t like as much as the first three, but he’s the undisputed authority. All four are books written mainly for historians, but I think they’re pretty readable (although I admit my perceptions may be skewed).
Oh, I forgot to add - I love McCullough’s series, but she is not a professor anywhere in history. She has great knowledge of the period, but she plays with it a lot - her books are definitely fiction.
To me it isn’t so much how our views may have changed in as much as that there’s been over a hundred years of archeology since Mommsen wrote his work. I can only speculate how much has been discovered in that time and what new information it reveals. It probably doesn’t have as much impact on our view of the BIG PICTURE as it does on our knowledge of the everyday lives of ordinary romans.