On my first trip to London (nearly 30 years ago), I remember being particularly awed by the Tower, as it just seemed so much more ancient than anything else I'd seen at the time.
Westminster Abbey impressed me more than I expected it to, as did St. Paul's.
As far as museums go, I'm quite fond of the British Museum and the Tate. But I always find the Natural History museum to be a thrill, less so since they've upgraded it but you can still see the penguin eggs Apsley Cherry-Gerard brought back from Scott's expedition. The V&A is worthwhile too, but pricey.
Kew Gardens is worth an excursion. I'd rank Greenwich as my favorite close-in day trip.
I've also taken a number of the walking tours offered by London Walks and found them uniformly entertaining and informative.
I also usually go to the theatre and stop in at Foyle's to browse (and, alas, buy) books, though the latter has gotten less entertaining now that it's been reformed to some sane level of organization.