I can't name a best district to venture but I don't really agree when mousquet says that you should focus on the main touristy things, it is for that reason that many people fails too understand and believe that the city is centered around the Eiffel tower and other monuments.
My advice, take a subway map and chose randomly a metro station.
The random part is very important.
Take a pen, close your eyes and write a point on the map.
The unexpected is the best thing of the travel.
