If you've only got a couple of days in London, then pick the 1-2 things from a couple of categories of things you like, and do them. Have an alternative if you either change your mind or the weather is bad - walking around outside in London when it's pissing down all the time like most of this week is no fun.

"You should try to see a show" - only if you like musical stage shows, as that's what they (mostly) are. They're certainly not for everyone, including me.

Museums - pick those that will appeal to you. The main museums are all very good at what they do, so if you like costume and culture go to the V&A; if you like old stuff, world art and ethnography and all the things the British Army could loot, the British Museum; if you like science and nature, the Science Museum and Natural History Museum; if you like castles, The Tower of London or Hampton Court (takes all day from central London); and so on.

Art galleries and exhibitions: Tate Britain for old stuff, Tate Modern for modern stuff, National Portrait Gallery for pictures of people (only go there if you like portraiture, it's all terribly worthy but a bit boring if you're not into portraiture as an art form), many other smaller galleries.

Drinking: there's everything from ultra-modern hipster mixology to pubs older than the USA, the only common element is that it won't be very cheap :-/ Personally I recommend trying to visit at least one old pub in centre for a pint or two.

Transport: use Oyster as described above. Don't take a taxi unless you can't walk very far and handle some stairs, and be prepared to pay a lot for a taxi to/from Heathrow. You can take the River Bus for a cheap boat tour of the Thames, similar to using the normal bus services instead of hop-on-off tourist bus services.

Shopping: Oxford Street has many of the big name brand shops and department stores, Camden has a market of smaller stalls (you might like to go around there and have lunch in the area), and overall there's no lack of shopping to be had in London. If you're doing serious (hundreds of pounds or more) shopping, check into the VAT (sales tax) refund scheme.

Music, clubbing, etc: pick your poison, London has many sorts, and goes on pretty late, later than the Tube so prepare for either a night bus, a walk, or a taxi home (except the Tube runs some routes all night on Friday and Saturday night). I wouldn't recommend trying to burn the candle at both ends with night life and day tourism.