One of my favorite neighborhoods to wander around is Montmartre. There's lots of stairs and twisty steep streets to wind your way up to the top, where it is crowned with a big white basilica.
The neighborhood wasn't always part of Paris proper, so it became a place of bars and cabarets outside the high taxes and rules of the city. Artist gathered here, and it grew into a fantastic art scene that is still apparent. It still harbors a red light district that gets a little sketchy at night, but is generally an ok place to take a stroll. The hip bars are starting to move in anyway.
Every October, they have a weekend-long wine festival to celebrate the harvest. Montmartre produces the only truly local Parisien wine.
There's also a great little fabric district at the bottom of the hill. There's a few really high-end shops and a whole lot of discount ones that you really have to wade through the piles to find a couple of treasures.