The best way to experience London is to get on the Tube (subway). You can buy an all day pass for around £10 and it will take you almost everywhere. There is also the option of sighseeing tours on an open top bus.
The weather in London and indeed UK is unpredicatable! Though we have had an exeptional summer, the weather is getting autumnal and you may find yourself altering from a cardigan to a coat.
Most of the iconic tourist attractions like Bukingham Palace, Hyde Park, Piccadilly, museums etc. are in the central of London. A good leaflet guide, which is available everywhere, a rugsack packed with water, a few munchies, money and a cardigan/coat is all you need for a days trip around.
For shopping look around Oxford Street (circus) for the big departmental stores like Selfridges, House of Fraiser, John Lewis and Debenhams. There are some nice places to eat around Soho if you fancy.
For big labels and designer goods, check out Harrods, Harvey Nichols and the Sloane St/Square area for big names like D&G, LV, Armani etc.
For culture, visit some of the best museums in the world like Tate Modern, British Museum, National Gallery etc. They are all located within the central London area and easily accessible via the Tube.
Canary Wharf is also worth a visit. It is a rapidly expanding area with high skyscrappers and beautiful landscapes. Whilst there, check out Greenwich and the O2. You can also take boat ride down the Thames river from here.
Westminister is the political capital of London and is a must. Here, you will also find the London Eye - UK's biggest tourist attraction and most definately worth a trip!
And finally, don't forget to vist the "desi" areas like Southall, Wembley, Brick Lane and Green St for flavours, taste and culture of the sub-continent!
Some good sites to help you plan your holiday;
http://www.visitlondon.com/http://www.londontown.com/http://www.tfl.gov.uk/These are some recommendations for London...