^ I could write a novel,but I'll try to keep it as brief as possible
As a first-time visitor, you definitely will want to spend some time in Old Montreal. It's architectually beautiful and vibrant and will certainly give you a great feel for the "tourist" side of the city. However - depending how much time you spend in the city - be sure to venture into the downtown propper (Victoria Square, Saint Catherine Strett) along with some of the more residential neighborhoods.
A couple of our favorites are right on the Metro's Blue Line. The first neighborhood we really like is Cote-Des-Neiges. This is genuine mixed income, multi-cultural neighborhood with a real vibrant street scene, particuarly in the 1/2 mile walk around the Metro station. You won't find a ton of real fancy or niche places to eat, but you will see some local and national chain retailers/restaurants that the locals gravitate to and enjoy. (Some of them are actually quite good). There is also an open air farmes market right across from the Metro station, which has a lot of good stuff!
Another neighborhood we enjoy is Outremont, a few stops East on the Blue Line. This is a bit more of an upscale neihborhood, but it has a really charming main street area with a bunch of local eateries and pubish type places.
Of course, you can't go wrong walking around the plateau, which generally follows the northeastern branch of the Orange Line and is the center of the French retail/dining scene. A lot of different things to see and try up there.
Closer to downtown, here are a couple of restaurants we always make a point to stop at; There is the Deville Dinerbar on Stanley Street. The concept is basically an upscale diner, but the food is high quality and atmosphere is comfortable.
One other place my husband and I both love is the Bar-B-Barn on Guy Street on the west end of downtown, not far from the Bell Centre. Simply put, the place is a dive. It's been around forever and is very popular with the locals, particularly the anglophone community. But the food - basically chicken and ribs - is tremendous, the prices are reasonable, and the staff is extremely friendly and very welcoming to US tourists. In fact, although we've only been there 5 or 6 times, a couple of the waitstaff have already begun to recognize us.
Hope this helps! Feel free to PM me if you have any other questions as I could certainly give you a ton of other pointers/recommendations.