Hi, Spencer,
Watch out for those Boxing Day sales! Most of the big shops have huge sales on Boxing Day, as they try to unload all the merchandise that wasn't bought for Christmas. You may be able to find some discounted furniture for your university lodging.
The week between Christmas and Hogmanay used to be called the "Daft Days." The phrase was coined by the Scottish poet Robert Fergusson. That week (which actually continued in many cases until the 6th. of January) was a time of merriment, but in times of austerity (late 20th. century), much of industry closed down or went to a three day work week. So those days weren't as daft.
Hogmanay will be celebrated with events throughout Scotland, so you should choose a town or city which has entertainment which everyone likes. One of the highlights of my times in Scotland was seeing Macumba and Runrig at Bught Park, Inverness, on New Year's Eve, 1999. On the way back to our lodging, we stopped by an ATM to see if it would work. It did!
On January 1st., you'll not find much open, with the exception of some corner shops and the larger supermarkets (which will have limited hours). There are two main shopping malls in Dundee - the Overgate and the Wellgate. The Overgate is upscale; the Wellgate isn't. Look for concerts at the Caird Hall - there is often great music to be found there.
You'll probably have about seven hours of usable daylight each day, so be sure to get an early start on your daytime activities. There are lots of great restaurants in Dundee, as well as the best independent record store in Scotland. Mozza, an Italian restaurant and pizzeria, is very popular with students. They have special pre-5:00 p.m. rates for pizza. If you have access to a car, you'll be able to travel farther afield. If you're limited to public transportation, there are some wonderful towns and villages to visit along the East Coast. You could easily spend a day or more in Glasgow and Edinburgh.
Celtic Connections is starting in Glasgow on January 16th. It's one of the largest music festivals in the world, and runs through February 2nd. You may find something to everyone's liking there while the whole family is together, or maybe something for yourself after they've returned home.
Will you be attending the University of Dundee or Abertay University? You are going to love Dundee, and Scotland in general.
Could I possibly become a member of your family? :)
Best wishes for your studies, and for your time in Scotland.
Mike (Auchterless)