In FindMyPast, use Search > A-Z of Records > United Kingdom > Search "apprentice" to find 13 different record sets. Some cover the whole country for a limited time (such as Britain, Country Apprentices 1710-1808) while others are geographically limited (eg Parish Apprentices for Dorset). Note also that the FindMyPast blog has a post on How to Use Apprentice Records for Genealogy Research.
At Ancestry, use Search > Card Catalogue > United Kingdom > search "apprentice" to see three results. One of these (UK, Register of Duties Paid for Apprentices' Indentures, 1710-1811) seems remarkably similar to one recordset found on FindMyPast, but includes images that can be viewed if you have a subscription (or are at the Library).
A digital microfilm image of the Board of Stamps: Apprenticeship Books (IR 1) is available at the (UK) National Archives. (Remember Les warned that these files are BIG.) TNA also provides a brief guide to researching records of an apprentice or master.
If you were intrigued by Les's brief reference to the famous Livery Companies whose members were master tradesmen and freemen of the city of London, then you might like to explore these links.
- Livery Companies http://www.liverycompanies.info/a-z-list-of-companies/
- The Records of London's Livery Companies Online http://www.londonroll.org/home
- Other guilds http://www.cityandlivery.co.uk/the-city-explained/the-provincial-guilds-of.html