I've looked through the basic books of the setting: Planewalker's Handbook, In the Cage, etc. Sigil seems to be chock full of signs and postings, so it seems to suggest that most planars can read Common. The Hall of Records includes the "Bureau of Learning," which is described as "Referral service for mentors, sages, and teachers. Oversees libraries." There is almost certainly not a regular school system, but teachers exist, so I would assume all schools are private, and possibly that itinerant teachers are hired on for irregular terms. It might be interesting to come up with a variety of small institutions of learning that operate in Sigil, where players could have picked up their skills, or where a sage might reside and work.
Here are a couple to get us started!
This temple to the Muses, found near Zeus' soaring temple in the Lady's Ward, is one of the richest and best-known centers of learning in the Cage. Students must be worshipers of the Greek Powers, or at least their gods must have no problem with someone making sacrifices to the Greeks on a regular basis, for all students are expected to do so annually. The sacrifices are separate from the tuition, and neither is for the skinflint: oxen, wine, and priestly services each year run about 50gp per student for the sacrifices, and once this is done 100gp buys a year's education.
The teaching is done by priests of the Muses and Apollo, and follows the Greek style: first the trivium, training in grammar, logic, and rhetoric, followed by the quadrivium, teaching in arithmetic, geometry, music, and astronomy. Someone who has paid all necessary costs and attended diligently for 1 year should be able to list on their character sheet a 150gp piece of "equipment": "Trivium Training." This training instills literacy in a previously illiterate character, assuming any necessary expenditure of skill points should this be required. (The student will be taking basic reading and writing classes with the young children of rich Sigilians.) Further years of trivium training, once the character is literate, give one of three bonuses: a +1 "education bonus" to Diplomacy, Perform (Public Speaking), or Decipher Script. The student can choose which bonus is granted in a year of training. Three years of trivium study concludes the coursework at this level, which is sufficient to read and write at the level expected of a competent Sigilian. Having started at age 6 or 7, the student is now 10 or 11 and will often squire out, apprentice into the family business, or otherwise start their vocational training. Students who complete the trivium are allowed to wear a braided yellow and white cord, although few do.
For students who wish to continue to higher education, the next four years give "Quadrivium Training," +1 education bonuses to either Appraise, one musical Perform skill (chosen by the student, not repeatable; more serious musical training is required to progress further), Knowledge (the Planes), or Knowledge (architecture and engineering).
Students who complete the full 8-year course from basic literacy through the full quadrivium (7 years, if already literate upon arrival) are graduated in a formal ceremony and are allowed to wear a blue and white braided cord which is recognized by fellow alumni in Sigil. Now age 14 or 15, they have often completed additional tutoring in skills expected of high society members such as Knowledge (Nobility and royalty), and spent skill points on these other areas.
Advanced study at the Museum is focused on the areas overseen by the Muses particularly: poetry of many sorts, history, dance, and astronomy. Advanced study costs 500gp per year plus the annual sacrifice, and provides a +2 education bonus to the skill studied. Astronomy training manifests as bonuses on skills directly related to Knowledge (the Planes) or, with the right focus of study, Knowledge (Arcana). A student who has received their education elsewhere may apply for advanced study if they pass a DC 20 check in the skill to be taught.
Zraxus' School of Languages and Literacy
Few people expect a Xaositect to be a good teacher, and fewer expect a school in the Hive to last very long, but Zraxus spent his younger years adventuring up and down the chaotic side of the Great Ring, from Arborea to the Abyss, and you're a fool if you think gray hairs on a sorceror mean weakness. After a few decades jaunting about and making a good pile of jink, Zraxus could have retired just about anywhere in Sigil, but (who knows how a Xaositect ever comes to a decision) he decided he was going to found a school that could do some good for kids that needed it the most. He bought a ramshackle old apartment building in the Hive from its slum lord, parked himself in there, and cleared a mess of thugs, bub-sellers, and worse out of the ground floor of rooms -- by himself, which meant that word got around. He took over the rents from the upper floors, and he's known as one of the more lackadaisical landlords: stuff might go a long time without being fixed, but then he might send craftsmen through to install fireplaces and chimneys in every fifth room. Tenants put up with it because he never worries about payment coming a little late, and even a missed month probably won't jog his memory until you've piled up quite a few.
That would make him of little note in the Hive, if it weren't for what he did with that ground floor. Zraxus was always fascinated with how chaotic minds managed to communicate. After all, you'd think language is full of patterns and rules that have to be agreed to before anything can get said, and somehow those patterns would have to be conveyed. And yet, eladrins speak Celestial just fine, and Abyssal seems to be somehow burned into the mind of nearly every demon. Zraxus wants to know how this works, and if you ever wanted to find a place where lots of different, chaos-touched minds were jammed together in one place and needed to comunicate somehow without tearing each other to pieces, it's Sigil you'd go to to watch them do it. So he set up his school, and declared that it was absolutely free to any Hiver kid that wanted to learn.
A day at the school is as unstructured as you might expect. Hiver kids come wandering in and stay a few hours, practicing their letters or conversational skills. Zraxus splits them up into rooms based on a current language of study, so experienced students help teach newer ones. He gives assignments somewhat haphazardly but always seems to be able to keep students busy. Around peak, a basic lunch is served, usually bread and a soup of some root vegetable or whatever could be found cheap in the markets nearby. Zraxus won't feed random beggars, but kids of the right age do tend to show up at the school just before lunch and bug out just after. He tends to dote on tiefling kids, especially those with clearly chaotic ancestry. Zraxus will also, at random intervals, apply his babble ability to students, much to the amusement of their friends, while he listens interestedly to the results.
Zraxus teaches Abyssal, Celestial, Dwarven, Elven, Giant, Infernal, Sylvan, and Undercommon, including literacy in all of these, with Dwarven, Elven and Infernal mainly being studied for their alphabets; however, Celestial and Infernal make interesting comparison cases for Abyssal. The texts that he has, which students are often set to copying for practice, are thick tomes and scrolls, informative in their own right. In addition to languages, he'll tell stories about his experiences adventuring, and in order to explain this or that linguistic concept he often seems to end up going into detail about the planes themselves, his magic skills, or about their history or the inhabitants. A student who spends a few years at Zraxus' school has probably picked up a few points in the Knowledge specialties Arcana, Nature, History, Religion, and/or the Planes, as well as, of course, several languages.
The education is free; getting something useful out of it is just a matter of being motivated enough to show up regularly. Once a kid has a couple of languages and some skill points under his belt, he stands a fighting chance of getting a job and making his way out of the Hive, something the most dedicated kids are intensely focused on. Their parents too, which means that whenever some particularly noisome development threatens the area around Zraxus' school, the perpetrators tend to find themselves on the outs with a good chunk of the neighborhood and (so far) pack up shop for another block.
Zraxus himself lives in an apartment on the topmost floor of his own building (making him a rarity indeed among Hive landlords). He must have some fine equipment there from his adventuring days, not to mention a stash of whatever cash he had left over after buying the building. However, intrepid thieves tend to find the charred corpses of missing former colleagues when they try to get in, which the wise find a sufficient deterrent. Somewhat surprisingly, Zraxus has a Guvner wizard for a friend, by the name of Enfort, who stops by occasionally to talk linguistics and arcana with him. Zraxus sets his babble against the Guvner's comprehend languages ability, and they experiment on whether chaos or order is the stronger inherent force in language.