I have seen beautiful displays many times, but not on a quest. I lived in Fairbanks, Alaska for six years.
They do not occur every night and your chance of seeing them on a 1-or 2-night visit is very small, unless you time your visit to a period of high solar wind activity. This can be predicted a bit in advance ( maybe weeks) and the University of Alaska Geophysical Institute has a website with aurora forecasts.
Right now they are predicting high activity on August 1 for the auroral oval (high northern latitudes), but ironically it will not be dark enough to see them. In addition to the right solar activity, you need darkness (so the months of September through March are best) and clear, cloudless skies (so interior areas of North America in the US and Canada see them more frequently than Scandinavian countries with their maritime climate). You also may need to be awake in the middle of the night. On our last visit to Alaska in late August, 2012, people saw a nice display from our wilderness lodge in Denali National Park---but they happened to be up at 3 am.
If you really want to give it a try outside North America, Iceland Air offers Aurora tours to Iceland, including lodging and interesting activities as well as the potential to see the lights.