Byward Market is great for just hanging out and tooling around. A lot of great courtyards are hidden in little nooks and crannies in the Market too.
As for culture and what not, go to the Shanghai Restaurant in Chinatown for great food and awful karaoke.
If you're going to be skating on the canal, go to Dows Lake and get some beaver tails. Also nearby is Little Italy, which boasts many great little Italian restaurants and cafes. Some of the best coffees you could ask for. Also, DiRienzo's on Beech is reknowned for their deli sandwiches. Go there and you'll see what I mean.
I'm not if Winterlude is going to be over by then, but if it isn't you should check out the activities and ice sculptures in Confederation Park.
In terms of museums, there are many. Near the downtown core, there is the Museum of Nature, The War Museum, The National Gallery, and the Museum of Civilization in Hull. If you feel like trekking out some ways for museums, there is the Aviation Museum in Rockliffe and the Museum of Science and Tech just off of St. Laurent Boulevard. And I'm not sure if it's still around, but the Eco-Musee in Hull just off of Montcalm is interesting...