SPQR: A History of Ancient Rome
To call SPQR a history is deceptive. This isn't a chronicle of events and dates. The only way you'd know that there was a First Punic War, for example, is that the book talks about the Second. In truth, SPQR assumes that you already know the history, at least in outline.
Mary Beard is trying to do something harder. She's trying to get us to engage with the Romans, to see ourselves in them--or, in some cases, not--and to talk about Roman approaches to what it means to be a world-wide, urbanized civilization. As a result, she spends a lot of her time on the nebulous concept of Romanitas, "Roman-ness". Instead of dates, you'll find a discussion of institutions. Instead of an exclusive focus on The Lives of Great Men, you'll find a diligent effort to understand the everyday experience of Roman citizens (as well as Roman subjects and Roman enemies).
This approach can be brutally boring. SPQR is often argumentative, and it's consistently skeptical of the "Just-So Stories" approach to history, but boring it isn't. It's very well written without sacrificing academic rigor. I especially like the way Beard shows that ancient people were not stupid, and that they--or at least the intelligentsia--approached some of the same questions we'd have with some of the same attitudes. The only kvetch I have is that Beard is occasionally so skeptical that I was left wondering whether anything happened at all.
Don't read this as an intro. Do read it if you're trying to dig deeper.
If you are looking for an intro, one good place to start is Tom Holland's Rubicon. (The sequel is just out; it's on my list.) And, of course, this.