Paris has amazing food everywhere. You will quickly find that there are a few key shops for daily eats - a patisserie (pastry shop), a brasserie (a simple restaurant), a boulangerie (bread & pastry shop), and crêperies (specialty crêpes). Most places will have a formule or fixed menu. Avoid tourist areas (Ile St. Louis, Champ de Mars, Champs-Elysées, and the Latin Quarter are notorious) - they are overpriced and unsatisfying. Be sure to try other food nationalities in Paris. The city is full of incredible Lebanese, Asian and Italian restaurants. In general, there is a variety of places to eat –it’s up to you to discover your favorite croissant or crêpe!
If you are a vegetarian, then your best bet is to shop at the open air markets in your neighborhood and organic shops. There are also some quintessential Parisian streets such as rue Mouffetard in the 5th, which are lined with specialty cuisine shops spanning from fromageries to boulangeries to tiny fresh produce stores. Traditional restaurants usually have a vegetarian option on their menus. The options are rarely vegan but numerous vegetarian and vegan restaurants opened recently as the vegetarian and vegan lifestyles are more and more common amongst the French.
The supermarkets (supermarchés) in Paris are usually small but are numerous. You will easily find a supermarket close to you, wherever you are in Paris. Keep in mind that most of them close between 8:00 PM and 10:00 PM.
Open markets take place daily in different parts of the city, it is the best place to find fresh produce.
Paris also offers a wide variety of international cuisine stores (chinese, korean, arab, indian, british, italian, greek...). These can be a good option to buy basic cooking ingredients in bulk.