Ignore anyone who says they have a favourite cafÃ© or restaurant: only tourists have these and shockingly, they always happen to be the first serviceable place within 10 minutes of their hotel. In fact, ignore anyone who tells you that you "have to" or "should" see/do/experience anything, it's inevitably just bragging about what they've done instead of intended to be good advice.
I recommend you avoid the first 8 arrondissements completely as they are largely filled with clueless foreigners, overpriced bistrots and kitschy boutiques selling crap, not to mention you're having the same pre-packaged, trite vacation experience as every other rubber-necking tourist. There are obviously some items of interest in there but you won't know how to recognize them so don't bother trying. Do not go anywhere near the Champs Elysees or the Grands Magasins; these are just soulless duty-frees. In fact, don't buy anything in Paris unless you really can't get it elsewhere because it'll be 2x more expensive here than anywhere else. For the love of God don't waste your time visiting Gucci and Louis Vuitton like some Russian mobster asshole, Jesus, what the fuck.
Other than that, I recommend you not stop in the middle of the sidewalk in mid-stride to take a picture while I am walking right behind you. I really hate that.
On the do side, it's pretty simple. If you like art, the choice of museums is obvious. If you like monuments, likewise. If you like architecture, just walk around and keep your eyes open, or rent a vÃ©lib and go explore. If you like French gardens, the Tuileries, Luxembourg and the Jardin des Plantes; English: Parcs Monceau, Buttes-Chaumont and Montsouris. If you like shows, pick up copies of La Terrasse and the Pariscope to see what's playing. The OpÃ©ra Bastille is good, the Garnier less so. Go to the market and buy stinky, unpasteurized cheese you're not allowed to eat at home. Pair it with a tradition (not a baguette). Learn a bit of French and read www.paris.fr
to see what's going on in town in terms of festivals, expos, events, etc.