Gudauri is popular Georgian ski resort near Tbilisi (in around 100-120km or 70-75 miles).
It is better to go there by car and to have a photo or action camera with you, better both.
Gudauri is open during winter season. Besides traditional downhill skiing there is also HeliSki available, where you jump on slope from helicopter.
There are lots of hotels. Depending of what do you prefer you can choose either big and fashionable hotel.
or you can stay in small hostel.
Due to high popularity of this resort it would be better to make reservations beforehand.