Lots of awesome places to watch the Northern lights:
Lofoten islands, Norway
Abisko, Sweden
Ice Hotel, near Kiruna, Sweden
Iceland
Ilulissat, Greenland
North cape, Norway
Faroe islands

All those destinations easily reachable from Europe and not that expensive(except Greenland at only 4 flights/week)

The easiest is to fly to ARN, OSL or CPH and then take a connecting flight to TRS or KRN(which can also be reached by overnight train from ARN, way cheaper and funnier)