Neighborhoods to see
My favorite: Shoreditch — I wish I would have had more time to spend in Shoreditch and Hackney in general. Scott pointed me toward Hackney as a more untouched-by-tourists place to check out, and my favorite thing there was definitely the shopping on Redchurch Street. It's a hipster dream.
Other recommendations: City of London is the obvious one to not miss, but Primrose Hill is also lovely, as is Covent Garden, which we called home base during our trip.