What type of education do the high-born in Westeros have? For example I know they are literate but what else do they learn?
Most of the major houses of Westeros have a Maester contracted to either their castle or family. Although the Maester's primary role is to advise the Lord of the house, they often also train the family's children in addition to other duties. Maesters are trained at The Citadel in Oldtown and are considered to be among the most learned individuals in the world. Each Maester specializes in a variety of subjects, which are reflected in the linked chain worn around their robes (think of it as their resume). In addition to their specialization, all Maesters are trained in subjects such as medicine, healing, administration, diplomacy - generally the subjects needed when running a castle or army - which is one of the many reasons a Maester is a highly prized asset.
Once a noble house has made arrangements with their Maester as to the appropriate subjects for their children, the Maester will generally tutor the children as needed, using a scheduled curriculum. Each Maester will pick the curriculum for each child based on what role that child is destined to fulfill when they grow up. The schedule will vary, based on the Maester's other duties and availability. In times of peace, it is not uncommon for a noble child to have a daily tutoring session with the family/castle's Maester.
Lesser houses may or may not have a Maester depending on their status and/or finances. If a Maester cannot be contracted, then the Lord of the house will often find someone else to teach his children - perhaps a failed Maester or even a local scholar. In some cases, a child's training is augmented (or outright handled) by more than one tutor, with each individual tutor specializing in a single subject.
Common families, of course, do not have access to a Maester and must arrange the teaching of their children on their own. In most cases, a child is taught only what they need - usually the trade or skills of their parents.
In some special cases, the children of a common family will have access to a Maester via the noble family they serve. In such rare instances, the common family considers themselves to be extremely blessed, given that their children have at least partial access to the same training as a high-born child. In all such cases, said training is informal only, with the child sometimes sitting in on the high-born's study session - something that can only occur when the family's Lord is extremely generous.
It's more than just Maesters that teach the children of noble houses. Maesters teach things like heraldry, math, reading and these sorts of intellectual pursuits. Priests teach them about philosophy and religion. The garrison's master-at-arms teaches them about fighting arts. The lord of the house would typically teach them about strategy and warfare, as well as politics.
And these lessons do not end at adulthood either. See Tywin Lannister's instruction to all of his children. Or Eddard Stark's instruction to his. Moreover, children are often sent away for years to other houses to learn from them and make friends with other noble children. So any given house doesn't just teach their own children, but usually several others' as well.
The fact is that every noble child is given the best possible education from as many sources as possible in order to prepare them for the world. No expense is spared, and every opportunity is taken.
And just as interestingly, in the GoT universe, parents can be flexible enough to see that sometimes, a child doesn't fit the mold, and they go with whatever interests they do have. Examples of this include Arya Stark, Brienne of Tarth, and maybe Shireen Baratheon. A lot of other children are forced into positions that don't suit them, like Robyn Arryn, or Samwell Tarly.
And then there are some children, who should never have the positions they are born into.
Westeros doesn't have what you might call a school system. Each family basically takes the education of its children into its own hands. Traditionally, the education of a nobleman's household falls to the appointed Measter. We know little about this process, but it would seem a Maester is at liberty, more or less, to choose the curriculum they teach to the household. We've seen various Maseters teach reading and writing, but we've also seen Maester Luwin (House Stark's Maester) teach history, heraldry, geography, sums (basic arithmetic) and practical subjects such as running a household.
If you follow the books, consider the example of Griff in ADWD. He was trained by a Maester, a Septa, and a pair of knights.
Early on, Sansa and Arya had Septa Mordane. Her role was somewhere between tutor, governess and chapterone.