Place de la Bastille is where all aspects of Paris meet. The site of the famous prison is still a prime spot for demonstrations and you'll usually at least be accosted by a few petitioners even if there isn't and outright manif. It's where the teenagers are neither entirely dressed from The Kooples nor clad head to toe in shell suits, but are a bit goth instead. It's home to the modern opera house and endless independent bars and music venues. The edges are a bit grimy, but on a sunny day, the golden Spirit of Liberty statue glints as it appears to take off into the blue sky.
For me, though, the best bit is the port, which lies on the canal between the Place and the Seine. You can stroll through the park and along the quay next to all kinds of interesting boats and even watch the lock in operation before you pass under a grimy bridge and arrive at the river.
After that, if you walk along the river promenade, the towers of Notre-Dame will appear in the distance, and soon enough you will be back in chic, touristy Paris once again.