Two long city walks:

1 - The Southbank Walk - start at Westminster and walk across the Westminster Bridge. Walk along the river towards The London Eye (kids love it, and there are nice views). Keep walking and you'll pass more bridges, restaurants, art galleries, theatres, etc. You'll eventually end up in front of the Tate Modern. You can't miss it - giant smoke stack. It's an awesome inside space - pop in for free to take a look. Then cross the millennium bridge which will spit you out in front of St Paul's Cathedral.

That whole thing will take you 45 minutes to walk if you never stopped. I like it because it ticks a lot of tourist buckets: westminster abbey, parliment, big ben, the river, the southbank promenade, the tate and st paul's.

2 - Borough Market area - Borough Market is the Pike Place of London. :) Check open times before you head out. Places nearby to walk to: Shakespeare's Globe (they do a great tour), the Shard, London Bridge and Tower of London.

Favourite parks:
Hyde Park - largest park in central London - you can walk, bike, boat, etc.
Primrose Hill - (central) smaller, but pretty views and in charming neighbourhood
Hampstead Heath - (north central) huge and beautiful and/but you will definitely run into Ricky Gervais
Richmond Park - (south west London) enormous, tons of deer, great for biking

Favourite neighbourhoods for strolling, shopping, eating:
Covent Garden, Mayfair, Marylebone

Want to see a show?

Museums:
The Tate Modern
The V&A
Natural History Museum (which is very close to the V&A)
The British Museum

Restaurants:
There are too many excellent places to list! If you find yourself in a particular neighbourhood, let me know and I'll find some faves.

current faves:

Bancone, Covent Garden
Sushi Samba, Covent Garden