Do Your Homework
Do as much homework prior to the trip as possible. Start out by checking out the Family History Library guide for researchers: PREPARING TO VISIT THE LIBRARY. Bring all the research data that you may need to get off to a good start. You'll find that after a few days of searching one line, it's often nice to go for a change of pace and search something entirely different, so bring data on more than just one family line.
Weather and Clothes
We can't predict the weather, but December in Salt Lake City is usually cold and snowy, so bring warm clothes. Overshoes and a coat or winter jacket are a good idea as is a pair of gloves or mittens. An umbrella might also come in handy. Layer your clothes so that you'll be comfortable once you are inside the Library or hotel.