At MAN, stay airside—large terminals and security rechecks make staying near your gate best.
Take the train from Manchester Airport to Piccadilly (20 minutes, GBP 5). Walk 15 minutes north to the Northern Quarter. Spend 45 minutes walking Thomas Street, Stevenson Square, and Tib Street for street art, independent shops, and coffee. Head back to Piccadilly with 40 minutes spare. For a 5-hour layover, add 45 minutes in Spinningfields near the river.
Follow the Northern Quarter route, then walk to the Museum of Science and Industry (free, 20 minutes walk from Piccadilly). It covers Manchester's role in the Industrial Revolution and takes 1.5 hours. For lunch, Elnecot on Deansgate Locks does modern British food from GBP 13 to 22. Walk back along the canal towpath.
With 10 hours, add a visit to Old Trafford (Manchester United) or the Etihad Stadium (Manchester City): both offer tours for GBP 22 to 28 and take 1.5 hours. Old Trafford is 15 minutes from the city center by Metrolink. Alternatively, Salford Quays and the Lowry gallery (free) are 20 minutes by tram.