Vacation in Austria

I will be going on a business trip to Vienna in mid-January and will then add two weeks of vacation to that somewhere in the Alpes. So, I’m really looking forward to this but I'm still looking for a nice hotel, preferably something with unique atmosphere. Hope that I am not too late to find a room, any insider tips?

Also, I want to go skiing, maybe get an instructor (haven’t been skiing for quite some time).. good skiing schools you can recommend, or good areas to go?
Would you recommend to rent a car while there or is public transport generally good?

Thanks in advance!