The main places in London are:
FRC - where the GRO index books and census info are kept, also some will info.
http://www.familyrecords.gov.uk/frc/default.htmNational Archives - used to be the PRO, at Kew -
www.nationalarchives.gov.uk - lots of records including military records, etc.
LMA - London Metropolitan Archives, for parish records, workhouse records, school records, electoral records, etc, for the London area.
webpage hereFinally there's the Guildhall library which is where parish records for the 'square mile' in London and also records from the Livery companies etc can be found
webpage here.
There's also the British Library's newspaper collection reading rooms:
http://www.bl.uk/collections/newspapers.htmlIf I had to choose only one, I'd go for the LMA. (In fact, I'll be there this Saturday).