Heathrow has three different rail choices into London. The Piccadilly line is part of the Underground, the Elizabeth line is a cross-city railway, and Heathrow Express is a dedicated service to Paddington. Sharing an airport does not mean sharing a fare, route or ticket conditions.
Gatwick does not have a London Underground station. Its railway connection is National Rail: Southern and Gatwick Express serve Victoria, while Thameslink offers other central stations. Choose the arrival that reduces your onward journey instead of looking for a supposed Tube line extending all the way to the airport.
Stansted is linked to London by railway, not by an extension of the Underground. Stansted Express reaches Liverpool Street and offers an interchange at Tottenham Hale. From those stations you continue on the urban network according to your accommodation; choosing the right connection can prevent a detour with luggage.
Luton and London City are different airports with different rail connections to central London. Luton combines the DART shuttle with National Rail, while London City has a DLR station. Check the airport name and code on your booking before choosing accommodation, buying a ticket or selecting your departure station.