I did not grow up skiing but took it up as an adult. My oldest started skiing at age 4, my middle son at age 3 and my youngest at age 2. Because my oldest two ski (and now snowboard) faster than I do, I have had to up my game to keep up...I even do tiny jumps sometimes, since I have to be in terrain parks with them anyway.
Skiing is a great (but expensive) family sport. We are lucky in that there are local ski clubs for the kids starting as young as third grade & we live withing an hour of decent (for the East Coast) skiing. If you are taking up this sport more seriously, I thought I'd pass along what's needed/helpful (this was my email reply to a friend asking for info) We (grown-ups) have our own boots - that's the toughest part to fit and requires no maintenance. Oldest just got his own snowboard/bindings/boots/carrying bag with room to grow, so hopefully that will hold him for while (except boots), though there will be maintenance. We have poles too, but frankly never use ours, as they come with the skis. We lease for the season - it is about $110/kid & $135 or so per adult, plus a hefty deposit...but where we rent, if you leave the deposit on file it grows so you are essentially getting a person free the next year. It's cheaper than renting for a week at a ski resort and makes day trips more appealing b/c you are not spending time renting gear each time. This year we will get a pack of discounted day passes. We have a roof-top ski holder but it's no longer big enough with all of us + snowboard...so we will be moving to a Thule roof top box...we borrowed one last year & it was awesome...requires roof cross bars.
We also have our own helmets (& goggles, obviously). It pays to buy extra ski mittens for kids when you see them on sale and maybe an extra pair of goggles. It pays to wear proper ski socks - I prefer Smart Wool brand. It's best not to have the kind of girlie ski pants that fit inside the boot, lest you get something called "boot bite" - painful!
For the kids, I often actually buy their ski pants at Target, though I have sometimes found them on sale at various sports places...REI can be great, but shop early. For snowboarding gear, skate shops. Shop in-store & buy online once you know your sizes. If night skiing, you may need clear goggles (for helmets) in addition to darker ones for day... More on goggles: Ask for suggestions on what are most versatile - the light changes dramatically throughout the day and ones that are great for sun can make it hard to see when the light is flatter at the end of the day. There are goggles for people who wear glasses. Never wipe the inside of your goggles when they are foggy or wet...scratches will result.
Everyone will need:
long underwear (at least 2 pair...there is a brand of microfleece - Hot Chilies? very comfy/warm)
other top layer...only a tshirt of not too cold but something warmer if super cold
socks (wool is best)
(maybe sock liners & glove liners, though I don't personally wear them)
pants or bibs (snowboard pants often benefit from a belt)
jacket (I prefer jackets with vents in case you get hot)...if you will spring ski too, maybe under layer + over layer rather than one jacket
neck gator & or baklava (gator attached to head cover for under helmet - kids might like - I hate)
gloves or mittens - I prefer mittens, the more flexible the better - mine are Scott
shoes/boots that are grippy to wear to/from (though unnecessary for snowboarders, most likely)
a big duffle for family gear
maybe a day bag to leave at lodge
If kids will be in lessons, consider taking your own roll of blue painter tape & sharpie & labeling stuff the night before, as this can be a bottleneck
I carry a tube of arnica pellets in my pocket in case of injury or achy knees
walkie-talkies & instructions to kids on not abusing them...no one wants to hear them playing around on the radio. pick a less-busy channel & have instructions about leaving them on. Get rechargeable.
other helpful info
Drink water before/during/after...expect to have a horrible headache for one day if at high altitude - drinking water helps - drinking alcohol hurts
Kids can sleep in long underwear
put socks on first, then long underwear
have a plan on what to do if separated or lost
make sure the seam of long underwear is not hitting a tight part of boot
have a grab & go breakfast maybe
consider having granola bar or something in pocket in case starving/melting down kid & long lines...try to eat lunch off peak or not at main base lodge
packs of tissues & lip balm too
wear sunscreen on your face
either take off for awhile or leave your goggles on in gondola - less likely to fog
let your boots dry completely then wipe down. Don't leave them in the car overnight - cold plastic is no fun...sit them by a heater. Store them buckled off-season.
everyone will dump their gear all over when they walk in the door...either make them come back & pick it up or do it yourself without complaint...it will happen, so don't get upset.
establish a firm "carry your own" policy and learn how to properly/safely carry skis so you don't blind someone
teach your kids the rules of the mountain - not unlike driving rules
get your gear ready to go the night before...recharge radios.
Mornings can be hell getting out the door but then it's all worth it.