Shops: Nozbone and Vega near Canal St Martin (while you are here and if you want to eat good mexican for cheap, go to el Nopal it at 5 min walking from Vega)
Eating for cheap around Belleville: look out the street between Belleville m?tro and Pyr?nn?es m?tro, called rue de Belleville, there are plenty of good asian offers (one is specialized in dim sum and soup, and is best value for money you can find)
If you're into spicy chinese food (Sichuan style) go to Deux fois plus de piment, it's not too far from R?publique, I think it's m?tro Saint S?bastien Froissart.
If you don't know what to eat and you are close to st germain you can always go to rue de la huchette, there are tons of restaurants (not the best, not the worst) from all kind of food.
Be careful with montmartre, not really the best place to eat, because it's full of shit for tourists. But while your around there look out for the APC outlet (surplus), I'm sure your lady will appreciate.
Museum: you have to go to Louvre but choose what you want to see before going because it's too big to be visited in one day. My favourite museum is Mus?e d'Orsay (Famous painting museum in an old Train Station). Mus?e Pablo Picasso in le marais, really close to Supreme. The museum of national history is really cool, and there is a cool park too.
Parks: Check Bois de Vincennes, and Buttes chaumont, both are really cool places to take a walk on a sunny winter day.
You can buy a Navigo pass for one month, I don't recall how much is it, must be around 40e, but it's illimited for bus/tram/metro etc...
For Clubs, nightlife in Paris is not what it was, and I think they closed social club (I could be wrong tho). Look out for le Pompon if it's still opened.