I am Finnish although not from Lapland, but I've visited loads. Lapland for Finns is a bit of a Marmite thing, some love it and but it's not for everyone. I love it up there, winter is an experience I think because of the 'kaamos', the Arctic Night. It's not totally dark but like 3mum's aupair said it's a bit like twilight during the day. It's so beautiful, you can experience being in this remote corner of Europe where there is just wilderness as far as you can see, just trees and snow for miles and miles. It's not like ski resorts in the Alps where there are little villages and towns and people everywhere, in Lapland it's just Mother Nature! Some people get uncomfortable because it can look a bit bleak and desolate (trees and more trees, snow and more snow), others love it.
As for winter activities, there is everything. I go there to cross country ski, the tracks are really well maintained and you can stop at a 'kota' (a trad Lapp teepee) for a hot drink.
It will be cold, you really need proper clothing, merino wool base layers, down jackets, boots, the works.
Levi is great by the way and my favourite resort when with children, Yllas is also. PM me if you need any info