An interesting case study is Ireland - one of the only regions of Western Europe to escape Roman conquest and colonization, and, unlike the various Germanic kingdoms that sprung up out of the Empire's remains, did not in any way consider itself to be a successor to the Roman state. The best evidence of the early Irish perception of Rome comes from a literary text called In Cath Catharda, which is an Irish freestyle adaption of Lucian's Pharsalia which narrates the civil war between Julius Caesar and Gnaeus Pompey. The text is entirely ahistorical but presents an interesting insight into how the Roman state was perceived by a foreign people who knew about it.

The text makes it clear that the Irish audience were aware that Rome was a truly international state; Caesar's army is composed not only of Italians, but of warriors from across Europe and Africa - he marches against Pompey with a host of Franks, Saxons, Mauritanians, Gauls, Britons, Spaniards, Scandinavians and others. Of course this is not a real historical fact, but it demonstrates that Irish people were aware of the fact that the Roman Empire was cosmopolitan and stretched over most of the known world. Archaeological evidence demonstrates that Irish warriors served as mercenaries in the Roman military during the late Imperial period, so this may reflect some received cultural knowledge about the geographical expanse and cultural makeup of the empire.

In terms of the state itself, it appears as though its bureaucratic nature was basically incomprehensible to the early medieval Irish mind. Ireland was essentially a stateless society where political, social and economic hierarchies were basically the same thing; aristocrats and kings gained their power not through fixed titles, but through a system of clientship and patronage where the lord would rent cattle to individuals in exchange for their military service and supplying of food and goods to sustain him and his retinue. The Irish legal system strictly ranked various grades of king and commoners based on the number of clients they held or the amount of net wealth they possessed. This is exactly how the Roman political system is described in the text to an Irish audience:

Now Decanus was the appellation of the lowest of those ranks. He was chief of ten: he was the man who used to punish every theft and robbery and outrage that was committed amidst the City.

Centurio above the Decanus: chief of a hundred was that man.

Tribunus above the Centurio: chief of two hundred or three hundred was he.

Vicarius above the Tribunus: that man used to assume the function of the Comes when the Comes would go to converse with the king.

Comes above the Vicarius: chief of one city was he.

Tóissech ('chieftain') above the Comes: twelve cities were subject to him.

Patricius above the Tóissech; the righthand man of a King or Emperor; and this was his work, to pass judgments and decrees instead of the overlord when the king himself was weary.

King above the Patricius: three tribes were his domain.

Emperor above the King. Howbeit the Romans had not that rank until Julius Caesar took it by dint of his hand, as the tale hereafter will tell. The Emperor, now, was overlord of the world above everyone, and there was no one superior to him.

The text mentions that Rome was some sort of republic without a single king until Caesar (rightly) came along and becomes emperor, but still conceptualizes these republican offices as basically different grades of kingship that would have had parallels in contemporary Irish society. It's safe to say that during the time that this text was composed, it would have been almost impossible for an Irish person, especially an Irish aristocrat who would have been the texts' audience, to conceive of a political system that did not involve kingship in some way.

So in sum, In Cath Catharda presents us with two major points that demonstrate Irish knowledge of the Roman Empire during the early middle ages: they knew it was vast and incorporated peoples from across Europe and the Mediterranean, and that they had to conceptualize its bureaucratic state structure in the language of their own stateless yet hierarchical society because that form of political organization was alien and incomprehensible to them.

The answer to this question changes considerably depending on the exact era, place and social class you're interested in, but I'll try to give you some quick highlights, which the mods are of course free to remove if they don't meet their expectations re. historical detail and thoroughness. Bear in mind I'm speaking very generally.

Firstly, by the nature of our textual sources we know more about the education and historical self-understanding of the clergy and the nobility than we do about the experiences of any other segment of medieval society. As a rule, their conception of Roman antiquity was based on classical sources that formed the medieval Latin curriculum and patristic writings, but privileged the imperial period, particularly the reigns of Constantine and Justinian, who served as exemplars of Christian piety and good, divinely ordained governance, respectively. (Constantine's reputation as a champion of Christianity was aided by the so-called Donation of Constantine, a document forged in Rome around the 8th century to support the claim Constantine had bequeathed the Roman empire to the papacy in an act of translatio imperii, or the controlled, legitimate transfer of imperial power from one holder to another: a concept that strongly influenced medieval historical thinking. (The Donation of Constantine was famously debunked by Lorenzo Valla in the 15th century, at a moment when European learning's relationship with ancient Rome was undergoing profound changes that I'll try to sketch out later.) Justinian's legend as a wise ruler, on the other hand, was bolstered by the major role the Corpus iuris civilis played in the elaboration of medieval legal theory in the late 11th and 12th centuries.)

Medieval rulers' desire to position themselves as contemporary Constantines and Justinians can be seen in commemorative texts like Otto of Freising's Gesta Friderici, which compares the 12th-century Holy Roman Emperor Frederick I Barbarossa to Constantine directly.

Whereas high medieval rulers and clergymen, or in other words members of the medieval European establishment, to put it very crudely, appropriated the memory of the late Roman empire as a political, religious and moral beacon (and often contested each other's claims to it in the course of their ongoing struggle to define the relationship between secular and ecclesiastical power), the idealised memory of Republican Rome as an egalitarian society, or at least one that afforded dignity to all citizens, could be used to articulate critiques of traditional authority. In the 12th century alone we see examples of this trend in the Roman Commune, which took up SPQR (senatus populusque romanus, 'the Roman senate and people') as a rallying cry against papal overreach, and in Joannes Zonaras's unfavourable comparisons of the ruling Byzantine imperial family, the Komnenoi, to the more selfless and responsible, in his view, politicians described in histories of Republican Rome. (Whether medieval fans of the Republic were in fact taking at face value ancient sources that themselves used Republican nostalgia and antiquarianism to obscure the real standing of the senate in the imperial period or to intervene in contemporary political debates is a different question.)

Nevertheless, the image 'Rome' called up in the collective imagination of the people we're most familiar with, especially before Christianity's cultural monopoly began to relax in the late Middle Ages, was predominantly Christian and late imperial, largely because of how successfully Christian theology grafted itself onto the history of Rome. While Jerome and Augustine also fed into this tradition, it was actually Eusebius who married the Bible to Roman history via Christian typology in his Historia ecclesiastica, which portrays Constantine as a Christ-like figure that, by pacifying and Christianising the Roman realm, is preparing the world for the Second Coming and the end of history. Although most medieval people would acknowledge that outcome didn't come to pass, Eusebius's reading of Roman history as a journey toward Christian revelation culminating in Constantine's enlightened reign became a model for later historians, e.g. Bede, who transposed Eusebius's framework to medieval England. Roman history would often be interpreted through the lens of scriptural sources like the Book of Daniel as well.

As to which classical authors educated people would actually read in school, and therefore base their understanding of ancient Rome on, I confess it's been a long time since I studied this and I'm afraid I may get the specific lineup and the historical changes it underwent wrong, so I hope someone else can take up this part of the question. I can tell you that patristic authors like the aforementioned Jerome and Augustine were more widely read than 'pagan' authors, but Virgil was nevertheless very popular. Cicero's influence grew beyond the school subject of rhetoric in the late Middle Ages, specifically the fourteenth century, when the emerging class of literate lay clerks, scribes, etc. in Italy developed a new interest in Latin literature. This movement is known as Renaissance humanism and Petrarch was among its earliest and most influential members. (Again, I'm speaking in generalities, but it's actually hard to overstate Petrarch's influence on the course of Italian or indeed European literature.) Humanists began to take an interest in questions about antiquity that hadn't been asked often before but seem logical and even obvious to us: how did Romans speak, how did they build these monumental buildings, what did they believe, etc. etc. etc. Humanists studied Roman antiquity thoroughly and systematically, engaging in research activities we may class under philology and archaeology, and uncovered some hitherto unknown Latin texts, most notably Lucretius's De rerum natura, whose neglected manuscript Poggio Bracciolini 'discovered' in a Benedictine library and distributed widely. To illustrate the curiosity (and limitations) of these men, I'll give the example of the very serious and spirited debate they conducted in person, in letters and in treatises in the 15th century about whether Romans actually spoke Latin in their daily life or only used its literary form in writing; one of Leonardo Bruni's (I think) arguments against the notion of Latin being a vernacular language in antiquity held that Latin grammar was too complex for Roman women to master and therefore they couldn't have spoken it.

Humanism's examination of questions like this contributed to the historicisation of ancient Rome in the eyes of the literate classes, which for the first time began to grasp the cultural differences between late medieval Europe and ancient Rome, and to view the latter as an object of study rather than emulation. The roots of modern historical consciousness, as in the idea that culture changes over time, can be traced to this moment, at least in the European tradition.

I'll stop now, but there's a lot more to say on the subject and I only covered it from one angle. I hope other people can go into detail re. popular reception, material culture, change over time, etc. because the scope of your question is pretty wide and maybe calls for an overview of the many medieval classicisms we know about instead of a few very general comments. Nevertheless, hope my comment was useful.

Edit: Forgot to mention that Byzantines called themselves 'Romans' and retained a low but consistent level of interest in Roman antiquity, especially the post-Constantinian period, but Manuel Komnenos, a contemporary of Frederick I, was somewhat exceptional among Byzantine emperors in engaging in the same type of Roman antiquarian plays as his western colleagues (he would hate hearing them called that!) when he compared himself to Constantine and Justinian. His court responded to his ideological program by composing poems assuring him that he was greater than those Roman emperors had been, just as Constantinople, which was in the flower of its youth, was greater than 'the old granny' Rome was or had ever been, which is pretty telling of how educated Byzantines at the time saw their place in history (and how geopolitics informed their views). In general, educated Byzantines did not bother to learn Latin and explored the pre-Constantinian Roman past through Greek-language sources, with a focus on history and not on literature. But this more or less exhausts the knowledge I've retained about Roman antiquarianism in Byzantium, sorry.

I'm talking about the gaul-area case, it may not be relevant elsewhere.

During the early middle ages, the Roman legacy was still omnipresent in the society. Your local bishop/archbishop was probably still of gallo-roman stock, same as the great families from your area, perpetuating roman traditions. IIRC the sarcophagii(us?) of the Autin museum date from the VIth century and are in full late roman fashion.

The new germanic ruling class wasn't ignorant of what Rome was, as they long were either tribes/ethnies living near the border of the Roman empire, with a long history of trade and other diverse interactions with Rome, or Tribes who were actually in charge of ruling parts of the Roman empire, a delegation of power earned by their military services for Rome.

This ruling class sought, after the "end" of the Roman empire, to emulate and take inspiration from the exemple of Rome.

For exemple, while local nobles and political classes were subject to germanic laws (burgundian, visogothic or frankish ones), the common people were still tried according to the late roman laws, which survived till roughly the end of the VIth century. These germanic laws were also the result of the influence of Roman writting and legalistic culture on the germanic rulibg class trying to emulate Rome by codifying their customaries laws.

While people from the VIth-VIIth century mostly never knew the empire (and in the end, with the destructuration of the roman power in the Vth century, they never knew absolute roman rule), they were still culturaly closer to the late roman society than to a pure germanic one, with the newly set germanic ruling class seeking to emulate Rome by adopting social and political customs of the defunct roman empire.

Hi, this is a relatively common question that might be in the FAQ, not to discourage further comment though. I've answered it with regard to England here, the short abstract of which is that the pursuit of Roman legacy and ideals was central to early medieval European politics.