There would have been a fair degree of difference, largely depending on what the child was expected to go on to do with his life. Here's some information about the situation for the aristocracy in England; I can't really comment on how widely this applies elsewhere.

Eldest sons were, of course, groomed to inherit, and in the case of landed (rather than, say, merchant) families were expected to stay on the family's land and manage their estates. There were only a few professions that were considered suitable for the children of the nobility, and younger sons would typically go into either the church, the army or the navy; there were a small number of exceptions to this, perhaps most obviously public service of some sort such as diplomatic work and senior posts in the colonial service, and I've come across one example of a horse-mad younger aristocrat becoming the manager of the Prince of Wales's stud in the later years of the 19th century. But these were relatively few and far between.

Typically, education for such families would have consisted of a governess, located at the family's home and travelling with them, for younger children up to the age of maybe 7 or 8, teaching the basics of reading, writing and maths. Slightly older children would have more specialist, male, tutors, who inculcated the basics of a classical education (scripture, Latin, a little literature and history, and, for the more able children, perhaps a little Greek). In the 18th century the practice was for these tutors to accompany mid-to-late teenage children on a Grand Tour of the main cultural and artistic sites in Europe, which might last for several months or even a year, and typically took in places such as Paris, Florence, Venice and Rome. Travel and on-the-spot art appreciation would be supplemented by more lessons on the journey, though the impression is that relatively few tourers took this very seriously, and there are plenty of anecdotes about fairly elderly tutors being unable to maintain discipline among their young charges, who would take advantage of being away from home and school for the first time to gamble, drink and visit prostitutes.

All this would have been supplemented by social training, for instance dancing lessons with a dancing master; riding, certainly, would have been picked up from a very young age, but perhaps not formally taught by an outside specialist. Girls would have received tuition in genteel accomplishments such as music, drawing and perhaps French or German. Artistic accomplishments of this sort were considered much less necessary for boys, however.

Specialist training suited to the child's planned career could begin as early as the age of 11. Eldest children would not have been expected to have significant training in estate management or the economics of their estates (though they would have picked up a lot of information from observation, and many did take a significant interest in such things); they employed "factors" to handle the actual running of the land. Children destined for the navy would be sent to a training base such as HMS Britannia at Dartmouth and entered as midshipmen; they would receive basic training, then go to sea and receive practical experience, often on a ship commanded by a relative or a friend of their parents', from the age of 12 or 13. There were army equivalents, but because the training for soldiers was much less technical these would often not start until significantly later, and children destined for the army, together with the eldest child, might well be sent to a public boarding school at age 13-14, as much to socialise and discipline them as to receive a first-class education. Children earmarked for the church would go on from public school to either Oxford or Cambridge to get a degree in divinity.

I've done some work on the Third Marquis of Waterford, a notoriously wild early 19th century aristocrat who was orphaned in his early teens. He was an eldest son and so was expected ultimately to manage his family estates. He was tutored at home until the age of 14, then sent to Eton for three years, and from there went on at 17 to the most aristocratic of Oxford colleges, Christ Church. Because his parents were dead and he was ill-disciplined, he spent most of his undergraduate life living in London and drinking and gambling; Oxford required him to leave after less than a year. So there were limitations to what even the best-connected aristocrats could get away with, and the same would certainly have been true at public school - where the young Marquis was flogged so regularly that, after he left Eton, he made a return visit one night with several friends and stole the headmaster's flogging block from the school library, proudly displaying it as a trophy at the family seat, Curraghmore in Ireland. But even Waterford was said to have been able to quote appositely from the Latin "greats" at times, so some of his education clearly stuck.