Austria in winter

Ski resorts typically require you stay for several nights - if not a week. The first week of February is also the same week that all of Vienna goes skiing as it is "ski week" for schoolchildren. If you haven't booked yet, you might struggle to get something in a resort town like Seefeld or Serfaus.

It seems like you might want to focus on larger cities as it will be the dead of winter and you'll want to do indoor activities. Obvious choices would be Vienna and Salzburg. In Vienna, you'll want to go ice skating at the Rathaus (city hall).

we are thinking about staying in Salzburg and going for a day trip to a ski resort. Which one would you recommend us to go?
and also, is Hallstad a good place to go on winter?
thank you so much.

We used Salzburg as our gateway city spending three nights before taking a train to Fibetbrunn (2 hour) in the PilletseeTal area of the Kitzbuelheler region of the Tyrol:
http://www.kitzbueheler-alpen.com/en/pillerseetal/holiday-skiing-crosscountry.html

We XC skied but the valley is populated by numerous lift assisted Alpine resorts.

The Fiberbrunn Sportshotel Fontana is located at the base of ski lifts and had a Bobo children's program but your daughter would be on the older side of the children we saw in learning to ski programs.
http://www.kinder-club.info/en/about-bobo.html
http://www.sporthotel-fontana.at

The PilletseeTal area is served by free buses that can get you to any of the ski resorts in the five village region.

If you're in Salzburg, another option besides Hallstatt is St. Wolfgang. Charming small town on the lake with good hotel options & restaurants.

where is better to stay in Salzburg: old town or new?
Thank you all for the inputs. It was very helpful.

Salzburg: We stayed at the Hotel Mozart.
http://www.hotel-mozart.at
About a km from the train station and a short walk to the pedestrian only old town zone.

Austrian ski areas:
https://www.j2ski.com/snow_forecast/Austria/
http://www.onthesnow.com/austria/ski-resorts.html

From Salzburg, the Ski Amade group of resorts is nearby.
http://www.skiamade.com/en/winter/skiresorts/ski-amade
Includes Hochkoenig, Radstadt, Schladming, Gastein.

Another good ski resort near Salzburg is Obertauern. The town sits at a particularly high altitude, so the powder base on the lower slopes tends to be a little more consistent.

For sking in western Tirol:
St Anton am Arlberg. Expensive VIP area and many hills may be too difficult for young folks, but there are hills for all.
St. Anton, St. Christoph, St. Jakob, Pettneu, Schnann, Flirsch, Strengen
http://www.stantonamarlberg.com/en
Ischgl, Again many difficult hills, but ski all day without the same hill twice.
http://www.ischgl.com/en

Zugspitz Arena is more family orientated:
Ehrwald, Lermoos, Berwang / Bichlbach
http://www.bergbahnen-berwang.at/
http://www.zugspitzarena.com/en/activities/ski-board

These viliges are ski orientated in winter and don't have an old town per say and are not large like Saltzberg. Ski in the daytime, party in the evening...

By the way... if you're looking to combine skiing with a location that also offers a scenic old town, then I would suggest Kitzbühel. The skiing is justifiably famous, but unlike most winter resorts, the town center seems to have quite a bit of history (even if the old buildings are occupied primarily by hotels, bars, restaurants and high-end boutiques). For all the attention Salzburg gets, it's Altstadt is surprisingly small and monochromatic.

We once went to Austria in February, and skied the resorts around Innsbruck. Skiing is steep, icy and often not as much snow as in other parts of the world. It's more of an intermediate to advanced ski region.
But the beauty of Tirol is hard to beat anytime of the year. And you can easily get there by train from Munich--about 100 mi. The towns up and down the valley have an incredible number of hotels, bed and breakfasts and other accommodations.

Snow reports are coming in.
https://www.j2ski.com/snow_forecast/Austria/

Austrian Snow Reports - Snow Reports November 17, 2015 Open ski areas
reported by 9 Austrian Ski Resorts, including Hintertux, Kaprun,
Kitzbühel, Obergurgl-Hochgurgl and Sölden.

We are doing our first Austrian trip in February as well, though a little later. Here's how I have scheduled with the help of the Rick Steves' guide and this board:

Vienna - 3 nights

Salzburg - 2 nights

Fieberbrunn area of the Kitzbühel region - 4 nights for hiking. I was particularly taken by the area and some very attractive pricing for half-board at little B&B/pensione places.