Piccadilly line — the airport and museums line
Runs from Heathrow through South Kensington (museums), Knightsbridge (Harrods), Piccadilly Circus, Leicester Square, Covent Garden, King's Cross. If you only learn one line, learn this one.
Jubilee line — the modern, fast one
Westminster → London Bridge → Canary Wharf in about 12 minutes. Air-conditioned trains, fully step-free, and the line everyone uses to get from the South Bank to the West End in a hurry.
Circle and District lines — the riverside loop
They share track through Westminster, Embankment, Temple, Blackfriars, Mansion House, Cannon Street, Monument, Tower Hill. Effectively a sightseeing loop along the north bank of the Thames.
Central line — the shopping line
Oxford Circus, Bond Street, Tottenham Court Road, Holborn, St. Paul's in one straight run.
Lines you can safely skip on a short trip
Hammersmith & City, Metropolitan and Bakerloo cover important commuter ground but few headline sights. The new Elizabeth line isn't a Tube line technically but is the fastest way between Paddington, the City and Canary Wharf.