I remember my Latin professor telling us that rich roman wealth was counted in area of fields, grains etc

I am going to first off say that this is not correct, we do not have many examples of estate assessments but what we do have is given in money value. For example, when Pliny the Elder discusses the wealth of Crassus he gives it as two hundred million sesterces, not the size of the land, what it produced, or a list of properties. Your professor may have meant that Romans considered land and agricultural wealth to be the only "real" wealth and everything else is sordid, which is partially true. There is some moralistic literature such as Cato the Elder's On Agriculture to that effect, and reason to see a general tendency of people in the Roman world to consider landed wealth to be real stable and secure form (because it was) but the elite were still heavily invested in trade, urban property and manufactured goods.

I suppose my starting with that is a bit of a metaphor of how difficult it is to talk about the rural common people^1 without obliquely talking about the elites. The biggest difference between Medieval and ancient rural life is that we have a lot of sources for the former, particularly in the form of charters and legal documents, and very little for the latter. Granted, there is a lot of writing about "the countryside" but I am using quotes because it is largely in the form of paeans about the simple pleasures of country living which is about as useful to getting at the realities of rural life as an episode of "Green Acres" (there are also a handful of works on agriculture but they are concerned with elite estate management). Epigraphic evidence (that is, from inscriptions) can tell us a bit more, for example there is the so called "Maktar Harvester Stele", a semi autobiographical account from a man who was born a poor, landless rural laborer and worked his way up to being a magistrate. Obviously not a typical case, but it is one of a handful of examples where we actually have the authentic voice of someone who was once one of the the common people (if not even a bit below that!). There is also Egypt, where the dry desert climate makes for excellent preservation conditions, and we actually do have something a bit more analogous to the situation of High Medieval Europe in terms of sources. I am not aware of the sort of charter and land deed documents that is the Medievalist's bread and butter, but there is plenty of debts, legal disputes, and monetary transactions that produced documents that we can now read. A picture emerges from them of tightly interconnected communities with highly monetized economies (whether or not actual coins were changing hands, value was thought of in monetary terms) and a fair amount of political regulation. The problem here is that Egypt was thought of as quite unusual in the Roman world, so while we can say quite a bit about Roman Egypt, it is a bit more difficult to know what Roman Egypt can tell us about the rest of the empire.

Which leaves the final tool, archaeology, which is very powerful but also very dependent on uncertain factors like preservation, how much and what is being excavated, how those excavations are being recorded, etc. It is also agonizingly slow, for example from 2009-2014 the University of Pennsylvania a series of excavations called the Roman Peasant Project which only last year received a full publication, and that is a pretty good scenario. That said, the results are fascinating, and it makes an argument for Roman “peasant villages” that were semi-autonomous and materially egalitarian found in certain Medieval cases (also a pretty good and varied diet, with plenty of beans and pulses). But again the problem of representativeness: the excavations were done in the relatively marginal land of southern Tuscany, but how would this hold true of Campania? The intensive study of low status rural communities is gaining in popularity so hopefully one day we will be able to answer that question!

And speaking of variation, the empire was a massive place that lasted a long time which means rural North Africa and rural Britain need to be somewhat different conversations. Fairly recently a massive, multivolume survey called New Visions of the Countryside of Roman Britain that incorporates so called “grey literature” (in short, archaeological reports that are difficult to access or entirely unpublished) and there is a lot there for those who like very technical reports. For example, the road system had a major effect on settlement patterns and was a major determinant of where settlements were located. There was a great deal of continuity in settlement patterns from the immediate pre-Roman period, the authors even suggesting that in many ways the seeming explosion in building during the early Roman period was really a climax of the Late Pre-Roman Iron Age rather than a new period. Architectural forms and common material was a mix of pre-Roman and Roman styles, it is not true, as you sometimes see claimed, that the Roman empire didn’t “take” in Britain and life simply went on much as it always had, but also it was not indistinguishable from those villages in Tuscany.

So there is a great deal to say (those books really are very large) but it is difficult to confidently say much that holds true generally. Monetization is a big one: coins were present pretty much everywhere and even if they weren’t changing hands they were being used to describe value. It was also pretty well interconnected, even fairly small and out of the way areas were plugged into imperial networks (such as with pottery styles) even if distantly. There were very few places that could truly be called “cut off” and probably no hidden Brigadoon where nobody even knew what a Caesar was. Diet was pretty varied and nutritious, there is some reason to think that there were general declines in standards of living but that is a whole other topic. There were landless laborers who worked their way up to being town magistrates, there were small villages that did not want anything to do with that, and there were small, seasonally occupied huts in the middle of nowhere. There were rural shrines to intensely local deities, but some people went on pilgrimage to large religious centers that drew people from all across the empire. Some people left to join the army and sent remittances home, some people never left.

In short, Rome was a land of contrasts.

^1 I am avoiding the term "peasant" because even though I suspect you simply mean it as "rural non elite" it carries a lot of connotations regarding settlement patterns, land ownership, social relations, etc. Some authors do use the term "Roman peasant" and they do so because they deliberately want to emphasize those same things.