Well, you've gotta get started at a café. There are 3 of distinction (1 is just my favorite):
1. Café de Flore 6th Ar. : where people go to be seen.
2. Les Deux Magots : also St. Germain (of course), their desserts are phenomenal. (Of course, this is a city where they serve a pound slice of flan in any bakery you happen to pass by, but LDM has really delicious stuff.)
3. Le Select : my absolute favorite café, I think it is in the 5th, so we're still safely left bank here. The waiters are great and it is certainly more old guard than the previous two.
An honorable mention should be made for Hotel du Louvre, which happens to be right around the corner from A La Civette, which I'm sure you'll want to drop a few Euros in. Of note there, see if they still have Quay d'Orsai panatelas, which are as close to cigar heaven as you're likely to get, especially paired with an Armagnac, absolute perfection. Their café crémes are absolutely outstanding at HdL.
***
There is one restaurant that sticks in mind, it was phenomenal, but both times I went I was, how you say, drunk to my eyeballs, so I'll have to ask my wife what it was. You almost can't go wrong walking around St. Germain and picking a random place, though.
Finally, and on an unrelated note, do make sure to go to the Musée d'Orsay if you haven't been. The Manets alone are absolutely stunning. Picnic on the Grass and Olympia dropped me to my knees.
Can you tell I get a little excited about Paris? Greatest city in the world.
