NY is expensive, but you probably know that already!
Yoy need to pick somewhere fairly snow-sure, so highish. I would also go for somewhere with in house childcare - an operator like Esprit Ski might be a good idea. Not cheap, tho. They can provide everything - other operators are Ski Famille, Family Ski Company, Snowbizz.
The reason I said in-house childcare is because your fpur year old is unlikely to be up for skiing for more than a couple of hours - too tiring. You, on the other hand ( and 8 yr old) likely to need all-day ski school if it's your 1st time.
Sounds pricey tbh but depends where, what it includes and whether you need it all. Pick a resort with more going on than skiing in case you or dc aren't that keen. NY days are very short and can be cold.
I'd say Mark Warner tend to be on the expensive side - but they do have a very good reputation. There are lots of smaller independent companies who are very family friendly that might be worth a look - Ski Famille, Family Ski Company, Le Ski, Ski Esprit (although believe they're part of Inghams).
Don't forget the cost of having to buy ski wear, googles, ski mitts, ski socks, thermals etc.
That sounds a lot - as beginners you are unlikely to need a full area ski pass for a big area like val d'isere or similar. Better off coosing a small resort acordingly. I would strongly recommend a specialist family operator for yr first time.
No personal experience of MW but I think they are v £££.
We had a lovely holiday in Alpe D'huez in France with Crystal sat year when my two were 4 and 6. The crystal childcare was fantastic, fun and v caring. We just booked them into ski school in the mornings only, then used wrap around childcare until 2pm. The Crystal nannies would take the children to the ski lesson, collect them after, they then returned to the crèche and were given lunch; we then collected them at 2 pm. This was great as we felt that we spent plenty of time with them too, in the afternoons we either did some easy skiing on a beginner slope or just chilled/pottered about depending on how tired the boys were.
Are you beginners aso or just the children?
It was a brilliant holiday, both us and the boys loved it!
Tge ice skating may be helpful if they are doing cross country skiing, or for general fitness levels
I would maybe consider the folliwing resorts:
Lech/Oberlech Saas Fee (but choose your resort position carefully - eg. near main lifts) Flaine (ugly but very convenient and snow-sure) Corvara/colfosco Serfaus - but these latter two are not major package destinations.
I would avoid French mega-resorts at new year. They will be busy and £££