visiting Paris-July 2nd to July 11th

Please, someone recommend important places to visit in Paris that are not very touristic, but local hang out places, where to learn about the culture and at same time points of interest in a small window period of time, inexpensive, to take advantage of time?
Would also have interest in travel to Brussels and Luxembourg by train? Is it worth, or should I spend majority time in Paris Protected content ?
Appreciate all and any input! Thank you.