We visit a few different neighborhoods on my small group tours to Paris.  On your tour to Paris here are just 2 places that The Traveling Professor recommends visiting:  

Latin Quarter
Métro:  St-Michel,  Cardinal Lemoine, Cluny-La Sorbonne, Place Monge and more.
The area on the left bank that has traditionally been the home of students, classic cafés, and a lively village-like atmosphere.  It is a great walking neighborhood.  I would compare it to the East Village in New York City.  This is my most favorite place to visit and stay.

Marais
Métro:  St-Paul.
This is the Jewish and the gay section of Paris.  It is a revitalized neighborhood that has regained prominence in the arts and culture and in dining.  It is a charming walking neighborhood.  Eat a falafel at L’as du Fallafel on rue des Rosiers and have a pain au chocolate from one of the bakeries.  The Picasso Museum, Place des Vosges, and Carnavalet (history of Paris museum) are in the neighborhood.  3 hours.  Enjoy a full day if planning to visit the museums.

Here is one area I would skip on a tour of Paris:

Champs-Élysées
Métro:  Charles de Gaulle - Étoile or Franklin D. Roosevelt or Champs-Élysées Clemenceau.
Although it is well-known and a popular, I am ambivalent about it being a “best neighborhood” anymore.  It is still the scene of the Bastille Day parade and finish of the Tour de France, but it is barely French.  The glamour of this famous boulevard has given way to McDonald’s, GAP, Abercrombie & Fitch, and cheap trinket sellers.  

For more information on Paris, see my website, www.travelingprofessor.com or visit my FB page at: https://www.facebook.com/pages/The-Traveling-Professor/136094362898