A few favourite restaurants:

Les Philosophes, my favourite local bistro, Marais

Miznon - vegetarian Israeli street food, Marais

MaoDumpling - in the 3rd. order the eggplant

Big Love Trattoria - in the 3rd

Breizh Café - proper crêpes from Brittany, multiple locations

La Taverne De ZHAO, best noodles ever, multiple locations

The Terrace at Galleries Lafayette, casual rooftop dining, in the 9th

Treize au Jardin, amazing brunch/lunch, in the 6th, next to the park

On my list to check out:

Tekés Restaurant - Incredible!

Fellows - pasta!

Peppe Paris - best pizza in Paris?

Museums, etc:

Musée de Montmatre

Mémorial de la Shoah

Pinault Collection

Musée d’Orsay

Louvre

Jardin des Plantes

Musée Rodin

Musée Picasso

Bibliothèque Sainte-Geneviève

Entertainment:

Le Caveau de la Huchette

Chez Papa Jazz Club

Moulin Rouge

Paradis Latin