I've tried using my fianceé's tablet a few times, but find it difficult looking at the screen instead of my hand as I'm working.
Does anyone have any tips for becoming more comfortable so I can make better use of the tool?
Open up Photoshop/Gimp, pick a tiny brush and start drawing. Pick something complicated that has a lot of details and just start drawing. Before anything, you need to get comfortable to not looking at your hand, you need to learn to use it with at least a little precision without thinking about it.
Trying to draw something is probably the best way to practice the coordination. You can also pick something geometrical, like a house, which will teach you to draw a straight line. Actually learning to draw a straight line in any direction is very hard.
When I got my tablet, I also did this excercise, where you pick one direction, and you keep drawing one line over another, trying for them to be as close to each other as possible.
The end result will look something like this. After a while, try doing so in another direction, until you're comfortable drawing in every direction. The movement should not come from your wrist.
Unplug your mouse for a few days. Trust me it works :) It can be a pain at first, but you'd be surprised how fast you get used to it. Of course getting good with it at illustration/drawing takes a bit longer.
Also, make sure you download the latest driver from the tablet manufacturer as it may contain support for a lot of software on your OS.