Things to do when travelling in a new place:
Do use yelp to find local eateries and coffee shops; don't be afraid to strike up conversations with strangers. Sometimes you meet the greatest people that way.
Do walk outside, find a tall identifiable landmark and roam aimlessly when possible; don't forget your camera, you'll always regret when you don't have it.
Do watch the weather, it's crappy to head back early when you need another layer.
Do eat the local food; you might never get a chance to try reindeer sausage again!
Do buy some sort of momento and/or take lots of pictures. I like to buy an ornament everywhere we go. I write the year on the bottom or somewhere and then every year when we put up the tree we reminisce of all the great places we've been able to go and memories made.