I am going to be in Stockholm for a week in January and would love to see the Aurora. Any suggestions on where to go and how to get there?
Trains do run all the way north in Sweden, for instance up to Kiruna or to Narvik, Norway, and other stops along the route. See more about Sweden rail passes and advance-discount tickets at http://www.ricksteves.com/travel-tips/transportation/trains/sweden-rail-passes.
The aurora's are meteorological phenomenon. They don't always occur, especially further south (and Stockholm is pretty far south as things go). And you won't see them if there is cloud cover. You can probably find predictions online a few days before your trip.
You will have to go pretty far north. It will be a long train ride (Stockholm to Kiruna is more than 16 hours). I'd suggest you look for flights.
You are very unlikely to see the Northern Lights in Stockholm. I lived there for 2 years and never saw them.
I've heard that Tromsø, Norway is a good possibility.