I've lived in London, and there are a few things I do on every visit. I'd suggest:
1. Go to a London pub for "Sunday roast," which is the big early-afternoon meal that many eat on Sundays. Great way to get a feel for the local culture...you'll see families, kids, dogs, young couples, etc.
2. Go to the National Gallery. One of the best art galleries in the world.
3. Ride the tube.
4. If I'm with someone who's in London for the first time, I take them on a walking tour that goes through some of my favorite neighborhoods & past some great attractions. Starting at Harrods (Knightsbridge tube stop), walk to Hyde Park Corner, then Constitution Hill along Green Park past Buckingham Palace. Walk down The Mall to Charing Cross, then along The Strand (stop at the National Gallery if you like) and cut up through Covent Gardens. Turn left on Long Acre and head toward Leicester Square. Walk through Leicester Square to Piccadilly Circus. Walk down Piccaddilly (stop at Fortnum & Mason or The Ritz if you want tea). Once I get back to Hyde Park Corner, sometimes I'll walk up Park Lane & then cut through Kensington Gardens. Otherwise I'll go down Knightsbridge & Kensington Road. Eventually I cut down to Cromwell Road & walk back through South Kensington & Brompton Road back to Harrods. (Note: This walk can take at least a couple hours -- or even all day if you make stops.)