(Illustrated, clockwise from top: Tower Bridge, Buckingham Palace, Harrods.)
Art Galleries
London has two Tate galleries. The Tate Modern (Blackfriars, Southwark) on the south bank, as the name suggests, specializes in such modern art that I was once transfixed by a steel girder set diagonally between two walls: I honestly could not tell whether or not it was a fixture of the building or an exhibit... Tate Britain (Pimlico) houses the pre-Raphaelite and earlier 20th century collections. The National Gallery at Trafalgar Square (Charing Cross, Leicester Square) houses older stuff going back to the rennaissance: Raphael, Canaletto etc. Don't miss the smaller but unique National Portrait Gallery on Charing Cross Road next door. Buckingham Palace has a Royal Art Collection open to the public (see below). There's also a Serpentine Gallery in the centre of Hyde Park but I've never seen it. Special exhibitions aside, entry to most galleries is FREE.
Museums
The British Museum at Russel Square (Russel Square, Holborn) is the British Louvre: Chinese, Assyrian, Babylonian, Egyptian and all manner of ancient relics. The Victoria and Albert (V&A) (South Kensington) specializes in early modern artefacts including clothes. Kiddies adore the Natural History and adjacent Science Museums (South Kensington); the former is especially huge but tends to be a bit of a "zoo of the dead": full of Victorian stuffed animals (but dinosaurs too)... If you're into Egyptology the Tutankhamun exhibition is on at the Millennium Dome (North Greenwich) but costs about £20 entry. Most museums are still FREE.
History and Sights
Get the best view of town from the London Eye (Embankment, Waterloo, Westminster), an outsized ferris wheel on the south bank diagonally opposite Big Ben and the Houses of Parliament. Tower Bridge (often mistakenly called London Bridge) (Tower Hill) is the one that opens up. Despite the "ancient" seeming gothick stonework it's barely over 100 years old, which is pretty young for London... As well as grisly true-life torture chambers, the nearby Tower of London also houses the Crown Jewels collection, which are on open display and well worth a look if you stump up the entrance fee.
The best walk takes you from Green Park Tube across the park to Buckingham Palace. If this is shut, there's still the Queen's Art Gallery on the corner (lefthand side) which takes in visitors for a modest charge. Stroll down the Mall to Parliament Square (Westminster) where Big Ben marks the top end of the Houses of Parliament (oficially known as the Palace of Westminster) which has a free public viewing gallery (bring full ID and expect to be rigorously sniffed and searched). While you're in the area a poke around Westminster Abbey, venue of numerous royal coronations plus the most important royal burial ground in England (check floor plaques etc for who's where)... St James's Park, the littlest and prettiest of London's major parks is just off Parliament Square.
From Parliament Square go up Whitehall, the heart of British Government. Downing Street (the Prime Minister's official residence is at number 10) is behind gates about half way up on the lefthand side. Keep going and you'll find yourself in Trafalgar Square, the nearest thing London has to an official central point.
Shopping
For mid-range (often unique) clothes and other boutiques Covent Garden (Covent Garden, Leicester Square) has quite a choice and a central former flower market devoted to street entertainment and overpriced icecreams. Oxford Street is Britain's high street ~ a mile-long strip featuring numerous Pizza Expresses, Burger Kings and McDonalds and every major chainstore as well as the second and third largest department stores in the country: Selfridges and John Lewis . Tottenham Court Road (Tottenham Ct Rd, Goodge Street) is thee place for electronics bargains there are gerzillions of shops. But check the voltage! Ours is 220-240 ~ double the North American 110 volt standard; also British plugs have three pins (and it is not always safe to plug foreign two-pin appliances into bathroom shaver sockets). Charing Cross Road (Tottenahm Ct Rd, Leicester Square) always used to be the centre of the book trade; now more dispersed, though the tatty Foyles (formerly Britain's biggest book shop if not the world's) is still in place. It's also the most disorganized bookshop in the world with no rhyme nor reason as to where anything's to be found...
For luxury shopping Bond Street (Bond Street) intersects Oxford Street north-to-south and has virtually every designer name missing from Sloane Street (Knightsbridge, Sloane Square). Knightsbridge is, of course, best known for Harrods which, with a million square feet of selling space vyes with New York's Macy's as the world's largest shop. The food halls are fantastic and the cosmetics departments can supply virtually any perfume in the entire world, no matter how exclusive. Locals tend to gravitate to Harvey Nichols across the road which has an extensive ground floor full of cosmetics plus about five more floors of high fash and a top floor piano-tinkling restaurant.
Walk down Sloane Street to Sloane Square, home stomping ground of the hideous, braying 1980s pearls-and-twinset-clad Sloane Rangers who all seemed to be named Tilly, Hilly or something similarly ridiculous. The Peter Jones department store is renowned for housewares and home furnishings. The long Kings Road from Chelsea to the more vulgar Fulham (home of people who can't stump up any more than £3.5 million ($7 million) on a house, has loads of small boutiques (including (formerly) the 1960s Beatles' emporium Granny Takes a Trip and the Vivienne Westwood's 1970s Sex (later Seditionaries).
An alternative to Harrods' food halls is London's most grandiloquent grocer, Fortnum & Mason (Piccadilly, Green Park). It's a bit of a theme park in atmosphere (high on tourists; low on locals) but a great place to acquire that 1890s-style China tea caddy.
For toys, Hamleys of Regent Street (Oxford Circus, Picadilly Circus) always was until relatively recently the largest toyshop in the world.
Camden Town (Camden Town) is London's funkier, more alternative homeground... every other shop and stall seems to specialize in leather belts, Doc Martens and handcuffs... there are more extensive markets (specializing in fashion and nicknacks) than anywhere else (despite the recent Stables Market fire)... Camden is thee place to go for streetwear and clubwear...
... And I could go on and on...
But not without mentioning entertainment. The West End theatreland equals Broadway with over 12 million annual admissions to over 40 theatres... just about every major show in the world is here. Book well in advance for popular musicals. If it's not on Broadway it's probably here. And if it's not here, it's doubtless coming soon...
Guides:
The Aussie-Kiwi-South African-oriented TNT magazine is FREE from little metal boxes outside tube stations. Time Out (magazine) is pricey (well, it is to buy weekly if you live here) but well advisable for any visitor. Also the Time Out Guide to London guidebook is the one most locals would opt for. Time Out also do more specific restaurant, shopping, etc guides...
Money
Most international ATM cards work in most cashpoints across town. Traveller's cheques: pounds sterling, US dollars or Euros are all fine. Other currencies may be a little more inconvenient. There are zillions of bureaux de change, but they can be pricey. Post Offices are said to give the best exchange rate.
And that's about it. Tomorrow I'll put a bit about some more off-the-beaten-track places like funky Brixton and posh Primrose Hill but ciao for now...
One last PS re getting around. Bear in mind that the tube map really does NOT represent geographical reality (it's designed for easy viewing and planning of journies)... so Monument and Bank appear to be connected by a mile-long escalator (they are NOT one station by the way and interchange seems to take ten minutes) it is well worth looking up anywhere you want to go on a proper map, judging the nearest stations and working out which is most convenient. E.g. to get to Hamleys, about a third down Regent Street there is little point changing to the Central Line for Oxford Circus if a Picadilly line train won't get you faster to Picadilly Circus (hang on is this a good example? my brain is going dead...) look it up on a proper map or you might well spend longer changing trains and lines than just walking from somewhere a little further off.