In terms of places to stay, I don’t have any specific recommendations, but when you are looking at ads look for places in “Centre-Ville”(downtown), “Vieux-Port”(Old Port) and possibly Ville-Marie. If you can afford Westmount, that’s pretty close to stuff, too. The transit system here is great, though, so staying a little further out doesn’t mean you’re that far from all the action. I can get downtown from the Plateau in about 15-20 minutes on transit. If you feel like biking, getting around on the Bixibikes could be useful to you as well.
It’s earlier than what you’ve indicated, but if you want to avoid crazy crowds, avoid the weekend of the 11-13 of June too, since that’s the Formula 1 Grand Prix du Canada.
If you like fireworks, ever Saturday night from June to August (and a few times on other nights as well, unfortunately), there are some rather spectacular fireworks hosted at the amusement park La Ronde which can be seen from many vantage points in the city. They even close down the bridge to road traffic so that people can sit there to watch the fireworks. I say unfortunately only because I happen to live close enough to have my glassware and pots and pans rattle with every boom, and unless we get out of the house early, we are prisoners in our own neighbourhood since the roads become packed with gaping idiots sitting in lawn chairs in the middle of them. The fireworks are beautiful though. I’ve just put up with them for 4 years!
Architecturally, the McGill University campus is rather interesting, and the museums there are interesting. Or so I heard!
If you like beer, there is are several good micro/craft breweries in Montreal and a ton of terasses to enjoy them on in the summer.
If you get here before they move on to Québec City and if you can afford/get tickets at all, I do recommend the Cirque du Soleil’s new show, Totem. I saw it a couple of weeks ago and was really amazed. And the hoop dancer guy is hawt!
Once you know a little more of what you’re interested in, feel free to ask more questions!