George H. W. Bush had the broadest variety (everything but state government) but was never in a given position for very long.
Easier would be least qualified, which is Spiro Agnew.
Chester Arthur begs to differ.
He held some patronage posts in New York, most notably Collector of the Port of New York and Chairman of the New York Republican Party. He never served in a Cabinet, nor was he ever elected to any office. He was selected as Veep because he was a supporter of Ullyses S Grant. Went on to be a solid President when Garfield was assassinated.
It's interesting looking at the backgrounds of Veeps how qualified they usually were for the office.
Agnew and Arthur stand out for their relatively unimpressive resumes.
http://en.wikipedia.org/wiki/List_of_Vice_Presidents_of_the_United_States