If you want to do a lot of museuming, check the opening days and times. IIRC many museums are shut on Mondays and the Louvre is shut on Tuesdays, but you'd have to check that.
I love the Musee Picasso, if you're into Picasso, obviously.
For Notre Dame the best way of seeing it is either to go to a service or a concert - we went to a lovely, fairly inexpensive concert there one evening and it was really magical.
There are loads of lovely places to eat near the new Opera (Bastille). You can also eat very well in the restaurant at the Musee d'Orsay. The Louvre has three places to eat inside it, and I can thoroughly recommend Cafe Richlieu (in the Richlieu section)
Most of Paris is really very lovely, but the Champs Elysees is fairly depressingly touristy and full of McDonalds etc. Many people also complain about the fakery of Montmartre. Yes, it probably is fake, but it's so nicely done I don't really care. However, you won't get the best value eating there as it's mostly fairly ripoffy "Menu Touristique" places doing nothing much more sophisticated than chicken and chips. It is a long flog up the hill to Sacre Coeur (you can get a bus though) and I personally find I have to be in the right mood to appreciate its slightly kitsch splendour. But go for a drink and a wander around and it's really lovely.
The other tourist trap letdown is the Sex Museum - lots of fertility symbol statues, all taken out of their own culture and context, plus some trashy artwork. In my defence, we only went because the guidebook raved about it, but it was awful.
Some of the very best views to be had over Paris are IMO from the Centre Pompidou's balconies and large gallery windows. For walking around, the Islands in the Seine are lovely - apparently there is an amazing ice cream parlour on the Ile de Vincennes, but we didn't quite make it there.