Ancient literacy is an oft-discussed subject, and it's one that we have a lot of disagreement on without a lot of sources. Certainly the wealthy and educated were literate, but where exactly the level of literacy should be placed (and what constitutes "real" literacy) is not exactly clear. This is something /u/Astrogator knows more about than I do (not without reason, since while I only dabble in epigraphy now and then for evidence he actually does epigraphy), and luckily for us he's written some very good pieces on here about the subject, which I generally agree with. So we don't have to reinvent the wheel--here's a good, representative one, literally just taken from the first hit if you search "Rome literacy" in the reddit search function.

For Pompeii in particular (is there any particular reason why all of a sudden half a dozen questions about the Pompeian graffiti have appeared?) it may be worth considering the nature of the site. Pompeii was unusually well preserved, which means that all kinds of stuff that would have been destroyed simply by time remain. That includes writing. Moreover, the graffiti at Pompeii are not all the same. The ones most people know are the short, often very vulgar little notices about the prices of taverns or prostitutes, or little boasts about sexual conquests. Those only make up a relatively small amount of the textual record at Pompeii. Besides that we have electoral notices, bits of poetry, and a lot of the time no (or little) text at all, just crude doodles. Famously there are two graffiti at Pompeii that combine lines from Ovid and Propertius. The first (technically listed as two separate inscriptions in the CIL, but they're written by the same hand and are really the same graffito), on the Basilica, contains a couplet of Ovid (1.8.77-78) followed by one of Propertius (4.5.47-48). These two couplets are masterfully paired (the Ovid is probably modeled on the Propertius), and whoever wrote these was obviously at least familiar with their verse. Another graffito at Pompeii is Virgil (with a local spelling of Olyxis for Ulixi), Eclogue 8.70. Again, not the stuff of some yokel. There's one graffito, which also combines Propertius and Ovid (different hand than the other one), that's scrawled onto the wall of a pretty nice villa, the Casa degli Scienziati, in which Propertius and Ovid have been modified and combined to produce a new poem (with a rather nice little picture). Obviously these, or the other lines of poetry (in both Greek and Latin) throughout the city are not the products of total baboons, and the location of the last graffito is not unimportant. But then again, the relationship that the Roman population had with verse, which would have been largely orally recited, would have been somewhat different than what we're used to today, so it becomes somewhat difficult to figure out at what social level the people producing such graffiti were. Moreover, Pompeii was not a typical Roman town. It was a resort town, relatively sparsely populated during the winter season, and its population does not appear to have been particularly representative of larger cities like Rome, which was packed with itinerant day workers living rather dreadful existences. Perhaps we should not be surprised to see so much casual writing