It was said in the first reply, and lots of people seem to be assigning private schools as the source of all evil, but there is only one way to achieve equality.
You'd have to take everyone's child away at birth and raise them ia commune, where the opportunities are all equal. All new babies would need to be anonymized to prevent rich people from finding out who their children are.
To consider the issue of schooling, let's suppose that we do eliminate private schools, and make public school attendance mandatory. There would still be some public schools that are a lot better than others, and there would need to be some way of deciding whose children can go to these, and who can't. However attendance is decided, the rich will be more able to jump through the hoops than the poor and have an advantage still.
So suppose we manage to make all schools perfectly identical. Kids only spend eight hours a day there, and the other sixteen they spend with their parents. Private tutors and the like have already been mentioned, but other factors still play an enormous role, from the attitude the parents have on various matters, to the diet and sleep schedules the kids have. All of these are areas where rich parents can have an advantage. The only way to avoid inequality due to parental status is to completely abandon the idea of a family.
If you want to have a completely equal society, you need to ensure that all children are taken from their parents at birth, given completely random names, and never see their parents again. In order to minimize genetic bias, only people of government-approved DNA would be allowed to reproduce. Once children are born, they are taken to one of a number of government-run facilities across the country. Each of those facilities would be identical, and the only people allowed inside would be Tue kids and those running the facilities. A month-by-month curriculum would be decreed, and children would be grouped into classes based on their month of birth. Teachers would rotate to a random new class and location every week, and children would rotate to a new facility (with all peers going to different facilities) every month. Anything less than this is allowing inequality between children to arise.