I'm not from there, but I've been there a few times, so my suggestions are mostly from the 'tourist' point of view.
1. Head down to Old Montreal and the old port area and just wander around. The promenade down there is nice just to wander. You can then easily walk around the old town.
2. Go to the Olympic park. I did the guided tour which is pretty cool. If the weather is nice, try and go up to the top of the Tower. The views are great; however, if the weather is sucky don't do it (and you may not be able to, because they don't run the car to the top if the weather is bad).
3. Go to the Parc du Mont-Royal and walk up the to various look outs. Again, if the weather is nice you'll get stunning views across the city,. At the foot of the Parc you can then walk around McGill university if you want, it's a nice campus.
4. If you can spare a day or two, go to Quebec City. It's really pretty and definitely worth going there.
Hope you have a great time!