Many of the non-elite inhabitants of the city of Rome were immigrants from Italy and other parts of the Empire (especially, it seems, the eastern provinces). But even for those who were born and raised in the city, and never had motive or opportunity to travel beyond its hinterland, it would have been hard to avoid a sense that the Empire spanned many exotic places and peoples.
The population of Rome (as I discussed in another post) was remarkably multicultural, and included contingents (if we can believe the poet Martial) of Parthians, Germans, Dacians, Cappadocians, and many others. Although there a few ethic enclaves (Trastevere was famously diverse), most Roman neighbors were probably ethnically mixed, with people of different languages, cultures, and religions sharing the same apartment buildings. Migrants from the far corners of the Empire continued to worship their own gods, and all inhabitants of Rome would have been familiar with shrines of Isis, Serapis, and other eastern deities.
Even if they were dedicated to more familiar gods, the great temples of Rome were filled with exotic treasures - ranging from Greek statues and paintings to mammoth bones and the skins of Indian pythons. These museums of Empire could be visited by the curious. More prosaically, those with sufficient cash could purchase exotic goods - most notably spices from India and beyond - in the city's markets, knowing that their exorbitant price reflected the distance they had traveled.
The scale of Rome's dominions was also on display in the Colosseum beast hunts (described in detail in one of my old videos), which featured exotic animals from every corner of the provinces (and in the cases of tigers, giraffes, rhinoceroses, and others, far beyond the frontiers). Triumphs, though rare in the imperial era, also paraded the products and spoils of a far-flung Empire.
Romans could even see a map of the world (and the Empire in it), commissioned by Augustus' lieutenant Agrippa. Although we don't know what it looked it (or how the boundaries of the Empire were indicated), you can see one plausible reconstruction here. The famous Peutinger Table, sometimes thought to be a copy, suggests the level of detail that Agrippa's map may have shown.
A non-elite inhabitant of the city of Rome, in short, would likely be conscious of the scale and diversity of the Empire, without necessarily knowing much more than the fact that it was very large and diverse. Commoners in the Empire's other large cities would probably have a similar, if less vivid, sense. It is hard to gauge how much the rural majority knew. They certainly understood that they were part of the Empire (tax collectors ensured that), but how much they knew - or cared - about its scale and diversity is impossible to say.