Sorry to be late, but pubs in London that I'd recommend to an American, simply because you'll see nothing like them at home, include:
The Lamb in Lamb's Conduit Street, Bloomsbury. Simply wonderful largely unspoilt Edwardian pub. Close by on Calthorpe Street is the Pakenham Arms, which is less distinguished in appearance but sells a terrific variety of excellent beers.
The Blackfriar at the north end of Blackfriars Bridge. This was going to be demolished to make way for redevelopment in the 1960s but a massive campaign led by the then poet laureate saved it. It has an art nouveau interior of marble, mosaic and beaten copper. Go in the evening, it's always packed at lunchtime.
The Lamb and Flag in Rose Street, Covent Garden. A bit of a tourist trap this, but still worth a look as one of the older pubs still around.
Actually, Covent Garden has a lot of good boozers, look out for The Harp if you're there.
The Cheshire Cheese in Fleet Street is popular with visitors. It's very old (survived the fire of London in 1666, IIRC) and was patronised by Dr Johnson who wrote the first english dictionary. Not at all a bad pub, despite everything.
The Princess Louise, Holborn. Big, busy pub with attractive interior.
The Star, Belgrave Mews West, behind Belgrave Square. Very very very expensive area, but - amazingly - a proper local pub with reasonable prices and excellent (Fuller's) beer.