This is a very subjective list, with a bias towards things away from the tourist trail. However, don't ignore the major museums, notably the V&A and the British Museum, or the most important historic buildings such as St Paul's Cathedral and the Tower of London. Links from this page outside my site will open in a new browser.
Otherwise, go for areas around the river, and go for the open spaces such as Hyde Park and Hampstead Heath. The sheer amount of green in the centre of London is quite special, and contributes a lot to its distinctive feel.
If you're at all interested, try and take in a play or a concert. The magazine 'Time Out' is a very good source of information, and has a website although it doesn't contain comprehensive listings.
Do take a ride on a Routemaster (double-deck, open platform) bus, and if the weather's good take a boat trip along the Thames. Hampton Court and Kew Gardens are both Thames-side attractions in the suburbs, and both worth visiting if you want a change from life in the centre. As you wander round London, be aware of the names of the different areas
Covent Garden and St Katharine's Dock are attractive places to wander in central areas, though possibly more popular with visitors than with Londoners. The King's Road is also a place to watch people: both the punks and the Sloane Rangers had their origins around here.
The National Trust has several slightly unusual buildings around London, notably a modern house, 2 Willow Road, in Hampstead, which hasn't been open very long and was the home (and his own design) of the architect Erno Goldfinger, and a pub in Southwark, the George Inn.
A few museums that you probably haven't visited are the
You will doubtless have thought of Greenwich; there's lots to explore there - take a look at the Gypsy Moth as well as the Cutty Sark, and marvel that Sir Francis Chichester sailed around the world in such a tiny craft. On the other side of London, take a wander around Richmond, and look for the point in the park where you can see St Paul's Cathedral, ten miles away, on a clear day. Go to Hampstead and walk through the back streets, and perhaps take a walk along the Regents Canal.
Markets are always worth visiting - go to Brick Lane in the East End and Camden Market which is at Camden Town, and also Camden Passage which is actually in Islington. If you're around on a Sunday, Columbia Road in Bethnal Green has a flower market. And the meat market at Smithfields is always worth a wander around.
Two books to look out for (though both are almost certainly out of print) about unknown London are: