I actually did not like SPQR, though I concede it may be a better introduction to someone who otherwise knows nothing. I found it too informed by 21st century perspectives, too set on telling me what to think, and uninteresting for that reason.
But, I know the subject material better than the average reader, and there is no single older work I can think of that covers the same time period. Gibbon's Decline and Fall is really much broader than Rome: he covers the Empire all the way to shortly after 1453, and not just Rome as such, but the broader theater of Europe and the East. But it does not cover the Republic or the early Empire. Gibbon surely is guilty of some anti-religious bias (which he confines, for the most part, to a single chapter), and there are a few things we have learned through later archeological research - but not much that would be of interest to a general reader - and a few things he did just get plain wrong. But as a whole, the work still stands tremendously, and for a non-academic historian, there's nothing wrong with it - except that in this case, you're not getting the whole history of Rome.
I actually think its a great read for a casual reader, as long as they're looking for a history of the time period specified. It is long, not dry at all, and the style is very refreshing. But for the earlier history of Rome, I'd do what Gibbon did for his work: go to the primary sources. Read Livy, Plutarch, Caesar, and the like. Read the Romans as they presented themselves, without anyone else interpreting it for you, or putting any kind of lens in front of you. To be sure, all the Roman writers had biases too, and where history gets hazy, they are sometimes much more eager than a modern writer would be to plunge into myth - though they are usually willing to say so, and often present interpretations of what they think actually happened. To be sure, there will be a lot of things we never know, things there can never be any evidence for aside from stories. But if nothing else, it adds color.
I've also found that modern academics are far too skeptical for my taste, and too ready to take absence of evidence as evidence of absence; generally too ready to deem myth as having no basis in reality whatever, improbable stories as impossible, unlikely ones as highly improbable, and even entirely plausible ones as exaggerated or unlikely, if any historian has lately, in need of making a career and publishing papers, an axe to grind against it. Half of any modern academic history nowadays seems dedicated to sneering at the gullibility of earlier historians. To be fair, maybe this isn't a new trend, only an old one refined to new heights - I recall a passage in Decline and Fall where contemporary writers were shocked at the numbers involved in the Goth migration, and suddenly realized that maybe accounts of army size in the Persian War weren't so crazy after all.
I don't know. Maybe the best thing for someone unfamiliar altogether with it all is to read SPQR, then, if they are still interested, read Gibbon and the more popular and accessible primary sources.