We know about wars and leaders and philosophers and artists. But do we know much about their farmers or store owners or schools?
Depends somewhat on what "average" means, but the general answer would be relatively little. We know more or less what most people were doing daily, but we know significantly less about how most people would've responded to particular stimuli, especially political ones--there's a big question in scholarship right now about whether the people showing up at elections and contiones regularly were really representative of the city or whether they should be thought of as really rather small subsets of the urban population. For the most part most people in the Roman world lived rural lives on farmland that their either owned or rented. They lived daily just like farmers normally do the world over, and in the off season many of them migrated into the cities for temporary work on building projects and so forth--some of the immigrants never ended up leaving. That the rural plebs had a fair amount of ability to leave their fields during certain parts of the year should be shown by their large presence during elections--the Gracchi and others called on the rural plebs frequently, and they arrived for political actions in huge numbers. Given that we're told that during Tiberius' bid for reelection to the tribunate he was unable to call on the rural voters because it was the harvest season rural leisure was probably mostly seasonal, and we don't really know how indicative of the means of a dirt-poor Campanian or Apulian farmer the electoral turnout was.
The urban plebs are a bit of a different story. Rome was, already in the late Republic, the most diverse city in the world, and it's very hard to talk about "average" people--even for people whose economic status was very similar there might be all sorts of additional factors which we really don't know about. The particular people that you've mentioned, other than farmers, cannot really be called "average," believe it or not, at least not if we mean that they typified the lifestyle and economic activity of the majority of urban dwellers. The majority of urban inhabitants were neither shopkeepers nor schoolmasters (who would probably be slaves in most households) but destitute day-workers who roamed the city looking for work each day at sustenance-level wages (or below, if they had a family). They were largely semi-homeless (rent, which was extremely expensive in the city, was often paid daily), a floating population that migrated through the city. They had little free time, having to be constantly on the search for work, and less extra money, and they were malnourished and diseased--there are conflicting estimates of the mortality rate in the city, but they all agree that it was horrific, only barely being offset by new births (which had to be quite numerous to break even) and immigration from rural regions. How much real participation in politics they could afford to have is a troublesome question, and it's one place where our texts are simply lacking. There's an enormous amount of debate on the subject, with some arguing that the urban population at all levels was intimately familiar with the political activities of the city through rumor and the personal connections that were so deeply ingrained in Roman society. Others argue that in reality the ability to participate in political activities and the interest in them was probably quite restricted, and even though the urban poor participated it was probably mostly restricted to clients and people who, like shopkeepers, would afford to forego a day's work to go to an election or a contio. My own opinion is probably in the middle there somewhere, I'd be inclined to agree that given the way Roman society worked it's hard to believe that some degree of political interest was not shared among most of the urban population, but I'd also agree that interest or not it's simply impossible for the majority of the urban poor to have mustered out for political activities constantly (for all kinds of reasons). One way or another, though, the older model of the urban poor as a class of idlers supported by their patrons and the state's largess and freed from the need to work by slave labor has been thoroughly discarded by everyone who works on this sort of thing these days--the Roman poor were laborers who lived very hard, and often very short, lives.
Besides that there's not a whole lot we can say. Most laborers probably had families of some kind, to supplement income if for no other reason. Many were freedmen, and politically speaking all freedmen were urban dwellers since freedmen could only be enrolled in the urban tribes (though of course many freedmen did not live in the city at all--conversely, many members of the rural tribes actually lived in the city, the census was irregular in the late Republic and redistributions of the tribes were haphazard). Our epigraphy, though not at all a perfect source (certain types of people are more likely to set up memorial inscription), suggests that freedmen dominated trades like shopkeepers or skilled labor, the sort of work that was guaranteed steady work (because they owned their own workshops and shops)--in a (now outdated, but still relevant) survey Brunt found that nearly 60% of goldsmiths found in inscriptions were freedmen, and less than 10% were free born (the remainder were slaves). Freedmen probably also made up a large portion of the majority of the urban poor, the day-laborers--unfortunately we don't have any good textual evidence for these people, so we know very little about their demographics. Regardless of whether they were skilled workers or not the urban poor lived in squalid little insulae, with one lightless room containing the whole family (if you've ever been to Ostia and climbed to the top of one of the surviving insulae to look down into the rooms it's quite shocking how horrifying the conditions must have been--and the ones at Ostia are pretty nice), for which they often had to pay rent daily, thus often moving with whatever work they could find and probably on workless days going homeless. They had no facilities of any kind, save maybe a chamber pot--they relied on the public toilets and had to purchase their food as they had no facilities to cook it themselves (a poor Roman would be shocked to learn that "eating in" is the activity of the poor today!). Their food probably consisted mostly of bread, which was baked for them (often at communal ovens) with the flour that they bought or were provided (usually both--at no point were public grain distributions large enough to feed a family) and little else. On a good day they might be able to afford a little extra, and on a bad day they might simply have to do without entirely. Harrison suggests that the urban poor probably wasn't starving to death that often, but that the vast majority of the city was probably horribly malnourished. Besides that disease was widespread, regardless of social class. The mortality rate by disease is pretty debatable, but it's widely accepted that it was high enough that only the constant immigration into the city from rural regions and the provinces was enough to make up for it, and that paradoxically this very immigration caused diseases to spread more easily as new diseases were brought in and the immunity of the population was constantly thrown out of whack. If a worker missed a day's work he had to search for something else that would pay him or keep him fed. Perhaps he'd try to hang out with a patron or go down into the forum and hope that some politician was offering largess. Maybe he'd hope that a candidate for office was distributing cash to the divisores in his area--we're not sure how this worked exactly, though. He might be more or less forced to hang around--it seems likely that often when public disturbances and riots took place money might be handed out to the crowd, which was probably made up largely of people who didn't get work that day and couldn't figure out what to do now. Sallust even suggests that there were essentially private gangs of hired thugs, probably of the same sort of people, who could simply be enlisted--what exactly he's talking about is open to discussion.
But by and large we simply don't know. We know some aspects of the lives of Republican urban dwellers, and can guess about rural dwellers probably with some accuracy, but details are mostly lost on us. We're still trying to figure out what the breakdown of the urban poor was, since our texts really don't tell us very much there. Our information for the Principate is much better, but it's unclear how much of that can be brought back into the Republic--for example, the collegia, associations which we know existed in both the Principate and Republic, are mostly known from Imperial evidence, and it's quite clear that we simply cannot extrapolate back to reconstruct Republican collegia and their nature, as the nature of these associations changed quite radically under the emperors