Looking for a map and website of the London underground, bus and regional rail.

Transport for London website, with links to busses, trams, National Rail, etc., is https://tfl.gov.uk/

The Tube page is https://tfl.gov.uk/modes/tube/; Tube maps are at https://tfl.gov.uk/maps/track/tube.

This website https://tfl.gov.uk/plan-a-journey/ will answer all your questions (look at the tabs for Maps and Fares & payments).
And the colloquial term for the Underground is indeed »the Tube«. Don't ask for a »subway«. That's something completely different in London, but not in Glasgow.

For travel in London, get an Oyster Card - valid on the Underground and the buses. Touch the card in and out on the tube, but just on entry on the buses.

Regional trains operate on set fares and are not operated in zones as the Underground is. There are different companies operating regional rail depending on which direction you are heading.

Subway here is a dreadful sandwich shop chain.

Thank you everyone for the great information!
TomS

This may be overkill, but here is the map showing the lines where you can use your Oyster on National Rail services.

http://content.tfl.gov.uk/london-rail-and-tube-services-map.pdf

If you are in London for a week or month, a monthly pass will definitely save you money over Oyster PAYG, and a weekly pass may if you are travelling a lot. Weekly and monthly passes cover any consecutive 7 or 31 days, they aren't limited to calendar weeks or months as they are in some cities. A big advantage of weekly or monthly Travelcards is that you don't get charged extra to travel before 0930 on weekdays, as you do on Oyster PAYG.

To expand on Jennifer's comment, main line trains do have variable (non-zonal) fares if you are paying for individual journeys. But they are included in the standard zones if you get a weekly or monthly pass.

Philip,
Where do you buy Travel Cards?
TomS

Thank you.

Just remember that if you want to take advantage of the 2-4-1 offers for mNy London attractions, you need a paper ticket or Travelcard. We bought one-week Travelcards and used them a lot for the bus, for a discount on the river boats, and once on the Tube. We got two-4-one admission at several places, including the Audrey Hepburn photo exhibit at the National Portrait Gallery, the Churchill War Rooms, and others. You must buy the Travelcard at a train station ( not Tube) and you need a passport-sized photo. There are photo booths in most London train stations.