The local authority and county archives for the county in which the school was located are a good starting place. Trade directories and commercial directories that began to come out in the middle of the1800's will have the names of the various schools in the descriptions of each town and village. There can at times be quite a few private, fee-paying schools; small, private, grammar schools; church schools; and charity schools, etc. If you have an address for an ancestor, an old map can be of help to see the location of the nearest school.
If your ancestor went to one of the public/grammar schools, you have more of a chance of finding records, as they diligently kept records of their scholars. Many have been published.
Some private schools are quite well known, and these all have published registers of their pupils. If you are fortunate to have an ancestor going to one of these schools, you are likely to reap information such as parents names, addresses, age at starting the school and leaving; plus, some give a short biography of the pupils life after leaving school, along with their career, etc. Also, it is likely that with the large private schools, forefathers may have attended the school; so with the registers one can follow a family through generations.
For government-funded schools, realistically, from about 1870 onwards, the schools started to keep lists of their pupils, so it's reasonable not to expect much before then. The types of records available for the government-funded schools are school admission registers, school log books, and minutes. The school log book covers the every-day running of the school, incidents that occurred, names of the teachers, and school attendance, etc. — good for adding the little extras that make family history come alive.
Some schools were initiated by certain charities such as the City of London's livery companies; these records are held in the Guildhall Library in London. The London Metropolitan Archive has school log books and registers for the ones that survive. If your ancestor was in a workhouse, from 1834 modest education was given to workhouse children. Any records for these would be found in the parish Poor Law Union Records. Resources include, Colin Chapman's guide The Growth of British Education and Its Records or for London, Cliff Webb's An index of London Schools and their records.