Hi,
I lived in London for six years, some of my favourite places:
Regent Street / Oxford Street for shopping although Covent Garden / Neal Street / Floral Street are great too for less mainstream clothing.
Camden (Camden Lock) and Soho are also great to explore although be careful, like any big city be smart and don't make yourself an easy target.
You might be better with a travel card than an Oyster card if you're going to be using public transport all weekend.
Use the Bus rather than tube, you'll see more of London but takes a little more effort.
I was impressed with the design museum Tate Modern http://designmuseum.org/ you could walk up the south bank of the Thames along Southwark see Tower Bridge, Tower of London and the London Dungeon they're all pretty close.
Use the River Ferry rather than paying for tourist boat if you want to see the city from the river.
Trafalgar Square and The Strand are also great to walk through/along you could cut up through Regent Street and then Oxford Street.
For night life, the West End/Soho is probably you're best bet but again don't make yourself an easy target, head to Old Street/Islington if you're feeling trendy.
The tourist walks are good if you like that kind of thing, I'd recommend the Jack the Ripper Walk.
Have fun!
