There are literally hundreds of places in the Alps that would fit the bill, whether in France, Italy, Austria or Germany. Look at any of the tour operator websites (Nielsen, Crystal, Heidi, Inghams etc.) for an overview of what is available and at what budgets. There's no obligation to book with any of these companies, but you'll get lots of ideas to work with and there are also specialist ski holiday travel agants who can help you put together a holiday.
At the most expensive end of the market are the big famous resorts - Courchevel, anywhere in Switzerland, St Anton/Lech/Zurs, Kitzbuhel etc - but these would probably be wasted on beginners and the children will just be happy to ski anywhere. They won't know if you've paid £50 a day or £500 a day.
At the less expensive, but still acceptable, end of the market there are places like Kranjaska Gora in Slovenia, or Spindleruv Myln in Czech Rep.
Norway and Sweden are great for beginners and kids, can be very cold in January but OK in March. Skiing is more limited for the better skiers and the landscape is not mountainous at all compared with the Alps.
USA andCanada also have some excellent resorts, but not really so good for just a 7 day trip with children once the travel time and jet lag are factored in. Canada and in particular the US have also become very expensive for accommodation, lessons and lift tickets.