I used to go out with a Blue Badge guide, so I absorbed a lot second hand.
City churches like St Dunstan in the East ( although it's now just a ruin with an amazing garden tucked away) are lovely for a stroll. St Brides on Fleet Street ( whose spire was the inspiration for the modern tiered wedding cake) has a basement worth exploring. Clement Danes and St Mary le Strand are lovely inside. Christchurch Spitalsfield ( the Hawksmoor church mentioned upthread) is gorgeous, it's my favourite London church.
St Alphege in Greenwich and St Ann's Limehouse are also his - it is proposed that Hawksmoor, who was certainly no Christian, places his churches along ley lines, and that joined together on a map they form a pentagram...
There's also a mummy in the tower of St James Garlickhythe on Garlick Hill although the church is only open by special appointment.
St Paul's in Covent Garden is known as the Actors church and has gorgeous carvings inside by Grinling Gibbons. He also carved the organ case at St Paul's Cathedral.
You can do guided tours of the Barbican, which are fascinating and really opened my eyes to Brutalist architecture. The arboretum at the Barbican is gorgeous, although only open to the public in certain days.
Bunhill fields ( derives from Bone Hill) is a small but fascinating graveyard on City Rd. William Blake is buried there.
Crossbones graveyard in Redcross street SE1 is completely unprepossessing apart from the ribbons people tie to the gates. It is unconsecrated ground, and some 15000 prostitutes are buried there. It was closed in the 1850s.
Nunhead, Kensal Green and Highgate cemeteries are all great for an atmospheric stroll. Kensal Green has catacombs.
Dennis Severs House on Folgate street is a Georgian terrace ( you have to book) which has been recreated as an authentic 18th century house. It's very atmospheric.
The whole area around The guildhall is fascinating - you can visit most of the livery company halls when they have open days. The Guildhall museum is interesting and relatively unknown. Under the Guildhall is the fairly recently discovered Roman amphitheatre.
There is also a clockmakers museum ( all the clocks chime together at noon and it houses Mary Queen of Scots pocket watch which is shaped like a skull, a brilliant Memento Mori)
You can do tours of the Ghost stations of the underground, and of the great powerhouse pumping stations like Abbey Mills, which I believe is the only one still operational. Designed by Joseph Bazalgette, who had the foresight to make our sewers the size they are, he is also responsible for the design of many London bridges and also the Woolwich Ferry.
I can talk about this shit for hours