As first-timers with the family, I agree with the others that you should book with an established Tour Operator. Crystal, Heidi, Nielsen, Inghams are probably the largest and they organise everything for you. There are also numerous smaller, niche operators such as Ski Famile. Most offer hotel accommodation with half-board or B&B, although some have self-catering options. Some hotels are "apartment hotels" which give you the option of self-catering or using the hotel restaurant.
To get from the airport to your accommodation - if you're with a TO, a transfer coach will be provided. If you have organised the trip yourselves, there are usually shared taxi services or holiday shuttles from the local airports to the major resorts. E.g. Salzburg Airport has a page on the website listing all of the transfer services, and Lyon, Geneva have something similar.
Getting to the slopes - walk if you live close enough, or use a ski bus (some hotels offer their own minibuses) or use a taxi. Beware of ski-in/ski-out as this often requires you to actually be able to ski a steep run to get to or from the hotel - which as beginners you obviously can't (yet).
Lessons are a must - group lessons are fun and the children should also have group lessons with their peers. You'll probably learn faster in private lessons, but these can be expensive. All instructors in Austria will speak English. France and Italy can be less certain, but most schools will be able to provide an English-speaking instructor, and the TOs will have a contract with one of the schools (usually the one with the most English-speaking instructors).
You'll be able to hire skis, boots, poles and a helmet from any one of the numerous hire shops in the resort. The TO will have a recommended provider, and will also organise the lift passes for you. Unless you're staying close to the slopes, if possible, find a shop that lets you store your skis and boots overnight at the shop, so you don't have to carry them to and from the accommodation every day.
You need to provide everything else (jacket, trousers, fleece, neck warmer, ski thermals, ski gloves, ski socks, goggles, sunglasses etc.) which you can beg or borrow from friends, other families, work colleagues etc. or look on Vinted, ebay, charity shops etc.
You don't need the big, expensive resorts like St Anton, Courchevel, and other mega-resorts as you'll be paying for a lift pass that you'll hardly use. The TOs will indicate which resorts are beginner-friendly.
There is actually a separate skiing forum here on MN - and a helpful forum for skiers over at Snowheads.com