History: McCord museum downtown, Pointe-a-Cailleieres in Old Montreal. The latter is more classical, the former more suited to tourists looking to have a good time (and the history is more proximate in time)
Art: MBA (Musee des Beux Arts, downtown). Wonderful collection. For music and performing arts, check local free publications
Sports: The Canadiens suck this year. They're not worth the ticket price, unless you are a devoted fan/connaisseur. Not much else going on during the winter. But if you really, really want to, there are a couple of curling clubs in the city ...
Nightlife deserves three distinct categories
Early evening (food): I strongly suggest at least one visit to Schwartz (St-Laurent), the best smoked meat place in the world, as seen in National Geographic. Yes, the health department would look unkindly on the fauna, but it's an institution. And anyone who tells you that Reuben's is just as good is an idiot. As for food in general, the good old rule of thumb applies: follow the locals. Just stay out of places that are obviously for tourist, like all restaurants in Old Montreal. And at some point, you have to try the poutine (fries, gravy and cheese - order s amall, since your stomach lacks the training to tackle a large). I recommend the Rapido (St-Denis & Mont-Royal) for dinner (above mentioned delicacy) and a show (freaks on the street at 3 AM). If you are fealing more gastronomic, I recommend the "Pied de Cochon" (pig's foot, literally). Looks like an ordinary bistro, but fabulous cuisine (a bit pricy, and a bit showy - I doubt anyone will ever order the Romanee Conti from the menu).
Mid-evening (beverages): many clubs for many tastes. Bear in mind that the city is divided in two down the middle, and that venturing on the East side might require some half-assed french with plenty of apologies for your ignorance of their beautiful language (just say you're ashamed of your president, regardless of whether or not you are). Saint-Denis is rather safe; the Plateau is hip; and the farther east you go, the cheaper it gets (and not in a good way). If you want to stay among anglophones, you can try Crescent ($$$) or the West Island in general (might require a car). Just remember this key rule: morals are looser, bear is stronger.
Late Evening (strip clubs): Whatever you do, do not go to Pare. Yes, it's the most famous, but it's also the worst boob for the buck. Same goes for downtown in general. They cater to stuck up American tourists, most of them from Burlington and aged 18-21. What you want are the ones on the periphery. Outside the island is even better. Anything in Laval will provide you with an incomparable ethnological journey into the parallel economy and social stratification, social reproduction, and basic capitalism. Just be aware that boundaries exist, and that humourless bikers control all strip clubs in the province. If you stay within the lines, a good time will be had by all.
Oh yeah. There's probably also some scenery and parks and stuff. Not many people care about that.
But seriously, Mont-Royal, and the observatory, though kitch, is sort of nice. I also recommand driving through the narrow and uneven streets of upper Westmount, where houses are rarely bought and usually inherited. That's is how the other half (ok, 0.1%) lives.