Welcome!
Firstly, my advice for not-that-cold is that you need to stay dry as well as warm, as wet will mean cold very quickly. You don't need to go all-out with down jackets, etc. for adults especially if you are likely to need to help your children up, carry their gear etc. but you will want water-resistant or even waterproof outer clothing.
Check what you own already - and DO NOT wear cotton. My top recommendations for things to purchase for even the newest skiiers are: helmet and goggles, warm mittens, and ski-specific socks (ideally merino, unless you are allergic, as merino socks are so much less stinky and can often be worn and aired out alternate days)
Most people own synthetic thermals, fleece layers, sports clothing etc. so look through what you can repurpose for skiing. Land's End often has good sales on their skiwear as well.
When outfitting children it's most important to keep hands and feet warm and dry otherwise they turn into balls of misery. This is also true for adults, but we have more control over when we go inside to dry things. So packing extra gloves/mittens and considering hand warmers as a back up is important.
Regarding washing/not washing/how much to pack, I have done a week-long ski trip (hiring boots/skis) with just carry on luggage. Clothing wise I took my ski jacket + pants (worn on the plane), mittens, balaclava, two pairs of leggings (so I could rotate and air them out), 3 pairs of ski socks, 3 pairs of casual socks, three thermals that I could layer, a fleece, and ski jacket/pants, and underwear for each day but rotated between 3 sports bras. I run cold though and tend to liberally apply antiperspirant to my feet to avoid the dreaded "slightly warm and sweaty feet indoors and then frozen on the lift + slopes) so you could probably get away with one less thermal layer. I didn't do any laundry until I got home.