I agree with the South side of the river. Few people visit the South side of the river but it has some interesting things.
The original "clink" prison in there (where we get the name clink from), as is Southwark Cathedral (Shakespeares brother is buried there).
The milenium bridge is right next to Tate modern, plus near the rebuilt Globe theatre.
There is also the London Dungeon
http://www.thedungeons.com/en/london-dungeon/index.html
There is also a National Trust pub, the George Inn, the last galleried inn (they used to stage plays in the courtyard).
http://www.nationaltrust.org.uk/main/w-vh/w-visits/w-findaplace/w-georgeinn/
If you have seen the Film Shakespeare in Love, much of this is set on the south side of the Thames in Southwark.
North of the river was the City of London and it was a very controlled area, South of the Thames was where people went to get their "sins" (wine, women and song) and many of the original globe type theatres were there.
You can walk Eastwards along the river all the way down to Tower Bridge.