We took DD skiing last spring, aged 9. First skiing for us all.
For lessons here, dry slope effectively (astroturf carpet on a travellator not brushes) so not cold - we wore leggings, ski socks and regular tops for those. Given boots each time on arrival (and we were hiring boots at resort so no need to buy yet).
For the trip, we got DD salopettes in Lidl which were great. (In fact, she has 2 pairs as we had picked them up a couple of years ago, and forgotten it when we saw them again, due to bad snows here a few years back). We brought both pairs which was handy when 1 pair got quite wet 1 day - there was a dry pair for going out for dinner that night.
Lidl also had great thermals, top and legs sets, which were great. I had 3 pairs of those, and we bought 2 new "proper ski" thermal tops as well. Lidl ones worked as well as the expensive ones. And gloves as well.
DD already has a lot of tracksuits - old style tracksuit fabric and reasonably fitted shape. So we just brought about 5 pairs of bottoms, and 1 of her more "professional sportsperson" Canterbury ones (double layers, zip at ankle, much wider and looser style) - old ones were great for days, and sloppy pair for night and mooching around hotel.
MIL had bought DD a good ski jacket in a sale at the start of winter - she spends a lot of time outdoors all year so it gets well used.
I think the only thing we bought new, really, for the trip was a pair of snow boots (reduced in sports store at start of season so €20), 2 extra pairs of ski socks (they are lovely and warm for hiking boots otherwise too), a pair of gloves and the 2 "proper ski" thermal tops.
And brought her older thermals, salopettes, hiking boots as well as new snow boots (as did DH and I), long sleeved plain tops, tracksuits (tops and bottoms), togs for pool and crocs for pool/hotel, and PJs. And a couple of woolen hats, a neck fleece, a couple of pairs of wool gloves - that we have anyway for winter use.