For a first skiing holiday in Europe go to Austria or Italy. Look at the package deals on offer from Crystal or Inghams - and come back here if you have any questions.
As for budget, I just looked at a holiday with Crystal to a reasonably big resort in Austria, staying half-board (breakfast and evening meal) including flights and with transfers to and from the airport included. I'm assuming that there are six of you - 2 parents, 1 adult (the 20-year-old) and 3 children. I've looked at w/c 10th Feb. as I assume that the children are tied to half-term - other weeks would be less expensive.
A perfectly acceptable 3* hotel was a total of £8,698 (£1,450 p.p on average).
A better, 4* hotel £10,260 (£1,710 p.p). A really good 4-star £12,140 (£2,024 p.p).
To this you need to add (approx prices):
Ski & boot rental - €100 pp
Lift passes - €350 for adults, €175 for children
Ski school - €250 p.p for group lessons for a week - €180 per lesson for private lessons.
Lunch & drink on the mountain - €20 pp per day
Beer, cola, gluhwein, prosecco etc in the afternoon after skiing - €5-€7 per drink.
Easter is early in 2024 - w/c 23/3 or 30/3 would both be good weeks to ski as most resorts will still have good snow conditions. Again, ask on here about specific resorts.