My fave places for a visit are Camden Market and/or Covent Garden, on a Sunday. They have such a lovely atmosphere, it feels quite fairy-taley. Camden would be more alternative to Covent Garden's tidy middle-class so pick what best suits you. Camden does have one of the few goth pubs London has to offer, though for regular rock pubs the Intrepid Fox is rather well known and has moved in the last few years just off the beaten track but still close to the corner of Oxford Street and Tottenham Court Road.
That ain't a million miles away from Orcs Nest. That being an RPG/Board Game shop. The service hasn't been as famously bad is it used to be known for, "service with a grimace" rather than a smile - now they smile, briefly.
If you like art, the National Gallery is well placed close to Charing Cross station. From there it's a mere shuffle to Trafalgar Square for a gawd at Nelsons Column. Can keep on trotting from there towards Westminster Abbey, you'll pass Horseguards and Downing Street on the way. From a distance, the Abbey always looked to me like a hobby construction out of matchsticks.
If you're more drawn to modern art could hit up the Tate, though if you're not into it it could irritate the crap outta you the way it does me "
wait, what? This is art?" 
Another mosey could have you start at the British Museum. From there you can hit up Oxford Street and the Plaza for (what used to be) the flagship store for GW. Keep on going in that direction you'll see a statue everyone calls Eros but actually ain't. A left from there takes you down Regent Street if that toy shop Hamleys is a lure, else keep on going into Bond Street for more high class highstreet stores. If you're a Sherlock fan, could always jump on the tube to Baker Street, the address the fictional detective lived at has a museum and shop given over to him that's kinda fun.
Wherever you go though, you'll see a load more than a Londoner. Living there, a person takes for granted all the statues and architecture. I've always had "
huh, not sure I've seen that before" moments when I've played the tour guide to visiting friends.
And despite all the touring, one of the things they've enjoyed to get a taste of England is watching the telly, even the adverts apparently have a quality to them. Also, a trip to the supermarket, I'm assured the kind of foodstuffs on offer say something about a country. I don't know what though.
