Don't know of any skiing opportunities. In Tbilisi I usually stay at the Marriott downtown, good location and well appointed rooms (plush and formal) but not cheap. Just down the street there is a Marrriott Courtyard, but I've never stayed there.
If you want to stay a little bit away from downtown the Sheraton Metechi Palace is a good choice. Wide open atrium design and Scandinavan-type room decor.