When I was in London I did a lot of touristy things and architectural sight seeing, but some of the coolest buildings/stores/shopping we saw from just wandering through the streets. Being from the United States one of the most memorable things for me was eating in a traditional English pub. I also remember one restaurant in particular (not a pub) called Give Peas A Chance that was absolutely amazing. If you like hummus definitely check it out.
I also saw a production of Wicked at the Apollo Victoria Theatre which was awesome.