Ryoumin Classes

So, proper education in the Heian period was likely limited to people in the Ryoumin classes.

Of those classes, the Imperial (皇族, kouzoku) and Noble (官人, kanjin) classes definitely got full educations with schools and tutors and in subjects we would recognize today.

Koumin (公民, which were mostly farmer peasants) would not need any significant formal education. Any formal records the Koumin would have needed to keep (such as calendars, inventories, tax records, etc.) would have been taken care of by a lower status member of the Kanjin class who would've been assigned to oversee their farm lands.

Although the Shinabe (品部, "court professional") and Zakko (雑戸, "military professional") classes were considered lower status than the Koumin, they likely would have had a basic education in reading, writing and arithmetic. This is because they had fairly regular interactions with the Kanjin class and it would have been good for them to have some level of formal education. They were also what we would call merchants & artisans, which means they would've been running small family businesses and being literate enough to keep their own records would just be practical.

Senmin Classes

Once you get to the Senmin classes, limited or as needed education would probably have been the most common.

The Ryouko (陵戸, servants of the Kouzoku or guards of imperial tombs), Kanko (官戸, servants of government ministries), and Kenin (家人, retainers or vassals of kanjin families) classes would have lived lives focused on service to others and the families/ ministries they served would've made final decisions regarding the direction of their lives. However, they did have the privilege of being able to marry and have children and they could not be bought & sold at will.

If the Ryouko, Kanko, or Kenin were given an education, it would likely have been at the discretion of the families/ ministries they served. I'm certain all 3 classes would have been taught the specific responsibilities and manners required for their service roles, but anything past a basic education would have been highly unlikely. Especially if they worked at very high levels of the government, an inability (or limited ability) to read would have helped maintain discretion and prevent others from using them as sources of sensitive information.

The last two classes, Kunuhi (公奴婢, government-owned slave) and Shinuhi (私奴婢, privately-owned slave), would have been very unlikely to receive even a basic education. They can be bought & sold at will and were not allowed to marry or have children of their own. These slaves are often used as manual laborers and would not have needed to be literate to do their jobs. They would have always had an overseer of some sort telling them what to do.

If a Kunuhi or Shinuhi were to have even a basic education, it's likely a former owner made an effort to give them one. Whether this basic education made the Kunuhi/ Shinuhi more or less valuable on the slave market depends entirely on cultural norms as well as what sorts of tasks their potential owners would want them involved with.

Zuijin/ Zuishin (随身)

A Zuijin/ Zuishin is someone who is a bodyguard of an Imperial (皇族, kouzoku) or high-ranking Noble (官人, kanjin) official. According to the Japanese Wikipedia page on the subject, anyone with this rank was considered a "ryogenokan" (令外官), or government official whose rank is not specified in the administrative code of the ritsuryō, or social hierarchy. In other words, they literally exist outside the class system. The reason for this is because Zuijin/ Zuishin are very high-ranking people (官人, kanjin) behaving like very low-ranking people (家人, kenin).

You see, the Zuijin/ Zuishin of the early Heian period (794-858 CE) were all members of a government office of the Imperial Court known as Konoefu (近衛府). To be members of this office, they needed to be nobles (官人, kanjin) of no less than 5th rank, which means they had full access to the Imperial Palace. So, the earliest Zuijin/ Zuishin were aristocrats (官人, kanjin) guarding other aristocrats (官人, kanjin) or royals (皇族, kouzoku). As the Japanese Wikipedia page says, "due to the nature of their duties, [the Zuijin/ Zuishin] had close personal relationships with the lords they were responsible for guarding, and these relationships tended to turn into personal, lord-vassal [a.k.a. kenin (家人) style] relationships." This was especially true during the Fujiwara Regency period of government (858-1068 CE), as the Zuijin/ Zuishin realized that the Emperors were merely figureheads and would have been keen on showing loyalty to the Fujiwara families in order to maintain their own high-status positions.