Like any big city, Rome was both a single idea and infinitely splintered reality - each street part of a system, every neighborhood a building block of the metropolis. And as in any big city, the ways its inhabitants experienced part and whole must have varied. Unfortunately, despite all the vivid detail provided by the letters of Cicero and the epigrams of Martial and Juvenal, our literary sources really tell us only how Roman elites liked to think and talk about life in Rome - potentially a very different thing from lived experience. Archaeology brings us closer to the sweat and dirt, but often merely in the sense of establishing the essentials of the physical setting. Your question, in other words, ain't easy to answer. But I'll take a wild stab at it.

Rome was indeed big - by ancient standards, almost unimaginably so, to the point that it strained even the High Empire's ability to requisition resources and funnel them into Italy. But for its size, the city was quite dense. Most Romans famously lived in insulae - apartment blocks - three or four stories high, built cheek-by-jowl along every street and alley (building regulations stipulating space between apartments were blithely ignored). The five or so square miles eventually enclosed by the Aurelian Walls contained the great bulk of Rome's million inhabitants. Almost no part of the city was more than a few miles from any other part; and if you knew where you were going, you could probably walk almost anywhere in little more than an hour. Unless you want to count the litters occasionally hired by members of the elite, there was no public transportation (with the exception of construction carts, in fact, all wheeled traffic was banned from the streets for most of the day). But that didn't matter, since from any given neighborhood, a sufficiently determined pedestrian could walk almost anywhere worth going in a relatively short time.

Although we have no way of proving (save by analogy with early modern European cities), it is likely that most Romans, like most premodern urbanites, worked near their residences. Local ties were strong in Rome, as the existence of the 265 vici (neighborhood organizations, each with its own magistrates and public cult) suggests. (One might also point to the odd festival of the October Horse, during which representatives of two neighborhoods competed for the head of the eponymous animal.) It should be remembered, however, that the great festivals of the Roman calendar brought all Romans into the ritual centers of the city, as did the combats in the Colosseum and the races in the Circus Maximus. Although Romans probably spent most of their lives in their neighborhoods, they would have never have forgotten that they were residents of a single city, and could never have doubted that their city was the greatest on Earth.