Another vote for Saalbach, or it's twin Hinterglemm.
As the OP is only a second-time skier, I'd recommend using a tour operator to organise the trip just to make things as easy to book as possible. The TOs will also have booked flight capacity and reserved hotel rooms in advance, and trying to organise something independently might prove to be expensive and possibly impossible as much will already be sold out.
Shared chalets have almost disappeared from the market as the operators can no longer easily recruit staff from the UK, so a half-board hotel is the best bet. You'll meet other families in the bar, restaurant or in the ski cellar and group lessons at ski school are also a great way to find some ski buddies of the same level.
New Years Eve is a big event in Austria. The ski schools put on a skiing show which ends with a big firework display at around 19.00. The hotels will usually put on a 5- or 7-course meal, there are fireworks on the slopes at midnight and the party goes on in the nightclubs until the early hours.
Looking at Crystal (other TOs are available, and I have no connection to Crystal or any other company) and flying from Belfast on 28th Dec. there are a few hotels that would fit the bill, although an outdoor pool will limit the options.
Hotel Saalbacher Hof is in the very centre of the village in Saalbach, 5 minutes walk from the lifts, and has an indoor pool (and a popular nightclub in the cellar for the teenagers).
Hotel Das Neuhaus is about 100m closer to the lifts and has an indoor pool, spa area - and also a nightclub.
Hotel Kendler mentioned above is also in a good location, and my personal recommendation would be the all-inclusive ValSaa Alpinresort (which has an indoor/outdoor pool) but neither are offered by Crystal.
In Hinterglemm, the Alpine Karawanserai is close to the Reiterkogel lift, and has an outdoor pool. If money is no object, then the Alpine Palace in Hinterglemm has to be the top choice - directly next to the Reiterkogel lift and with an outdoor pool.
A smaller, less expensive resort that is popular with MNetters is Rauris - try searching for threads on MN.