I start at the front page and browse. I look for anything new or interesting. I have a friend who is a reporter for the paper, so I look to see if he made the front page and read his article.
Then I look through the A section which includes national, international, and business news and the editorial pages. I look at the letters to editor to see if I recognize anyone or see a topic that intrigues me. I have several friends who routinely send in letters and get printed once or twice a year each. So it is fun day, if I recognize a name.
I look through the Metro section to see if there is anything I should be aware. I skip the obits.
I look through the Style section. First i look to see if the DC housewives have done anything- cause yaknow I'm a housewife in DC and they do exactly what I do. (blech). I love Carolyn Hax and Ask Amy and I read the comics. I save suduko and the crossword for later.
I don't generally read the sports section. I gaze at the specialty section of the day (Food, housing, Local...) to see if there is anything interesting.