You need to test meat, wine, baguettes, cheese, crêpes, glaces Berthillon...
Neighbourhoods to ask for drinks and meals are le Marais, Mouffetard/Monge, and the Butte aux Cailles in the south, Oberkampf where the students drink and around the Canal in the 10th more in the north, Belleville in the east,
Also of interest: https://github.com/abelards/how_tos/blob/master/paris.md (things to do)
These are in my "extra" budget so they're more expensive than normal meal but still affordable.
- le Pied de cochon (means "pig's foot": yeah, this is pork-heavy) http://www.pieddecochon.com/carte-et-menus/
- le Louchebem (meaning the name of a butcher's slang) http://le-louchebem.pagesperso-orange.fr/france.html http://le-louchebem.pagesperso-orange.fr/specialites.html
- I like the Épicerie Montorgueil epicerie-montorgueil.fr
Plus all the Marais and jazz clubs.
- lesronchons.fr near Notre Dame and the Tour d'Argent (menu: ~50€ without drinks)
- go for a Berthillon ice-cream on the Île Saint-Louis
- "Chez Rene" (14 Boulevard Saint-Germain, 75005) a typically french bistro; nothing fancy but great value for money. Presidents Mitterrand and Chirac came there.
- l'Aubrac for Aubrac meat
- comptoirnippon.com near Montparnasse (japanese teppan-yaki, 25-75€)
- "Bofinger" (5-7 rue de la Bastille, 75004) near the place de la Bastille (handy if you want to go out for drinks around Rue de Lappe). Ask to be seated under "la coupole". Don't mix up this place with "le Petit Bofinger" which is literally across the road and is nothing special (though the food is ok).
Enjoy Paris!