As first-timers I think that you should let a Tour Operator do all of the work. Crystal, TUI, Inghams, Nielsen etc. The Christmas week is not a particularly busy time - people like to stay at home - so prices can be reasonable. The New Year week is far busier and therefore far more expensive.
Everything that you need will be organised by the TO. Obviously transport to and from the resort, but also ski and boot hire, ski school lessons, lift passes and usually a social programme of some sort too (horse sleigh rides, tobogganing, bowling) if that's your thing.
All you need to organise is ski clothing - underwear, fleece, trousers and jacket (go to Decathlon or Trespass for good options at reasonable prices), gloves, ski socks (proper ones, not tubes with snowmen on) and a warm hat. You can rent a helmet, but if you prefer your own then Decathlon or Aldi have perfectly safe ones for £30-£40. You must also organise your winter sports insurance.
You can't go wrong if you pick any of the mid-sized resorts (which is where the TOs go anyway) regardless of whether you go to France, Austria or Italy. For a cheaper option look at Andorra - Bulgaria is cheap for a reason (but there are people who have had a great time there too).
France - Les Arcs, La Rosiere ... others will have other suggestions.
Italy - La Thuile, Kronplatz, Canazei ...
Austria - Zell am See, Saalbach-Hinterglemm, Westendorf, Soll, Rauris, Wagrain, Maria Alm ...
Personally, I prefer the village atmosphere and the teaching approach in Austria rather than in France - Austrian group lessons are huge fun and will give you and the teenagers a social group for the week.
There is an old thread that you could look at >www.mumsnet.com/Talk/skiing/3893018-First-time-skiing-how-does-it-work
particularly the post from @TeetotalKoala about 9 posts down.