The most important piece of equipment (other than warm and comfortable clothing) are the ski boots. For very young children and /or beginners, I really like the comfort and practicality offered by a 1-buckle, rear entry boot, such as this black model to the right. They are very easy to put on, have a wide range of adjustment, and can be loosened easily to facilitate walking.
As soon as your child grows into the sport, though
I recommend looking into multi buckle, higher cuff boots, such as
this blue model further below.
A higher cuff promotes a balanced forward/aft stance, and helps avoid the “back seat position”, where the child uses almost exclusively the back of the cuff.
The big mistake…. is to try to get 2 years out of the same pair of boots . Skiing is expensive, and kids grow fast out of properly sized boots. The best way to mitigate this is not to buy a too-big boot which will impede progress and enjoyment the first year (and may still be outgrown the second year), but rather to look into leasing programs (most shops offer season leasing); return programs – for example the Wachusett Mountain ski shop offers 30% of the original purchase after 1 year on equipment purchased new.
Or think of used equipment - guess what: people with a child 1 year older than yours had the same problem you are having– last year. In general, children are not heavy or powerful enough to abuse or destroy boots in 1 or even 2 years of use, and reasonably-used ski boots (not 5-time hand me downs, though) are just as good as when they were new – and considerably cheaper.
Children skis are wonderful these days - there really are no bad models to learn on other than the expensive racing models which will be too stiff for a beginner.
Size matters here, look at chin-high or even top-of-the-chest high for very young children and total beginners, going to eye-level high at the most for advanced, older children.
Overall, if you err, err on the short side. I would rather see my student come to a lesson with skis a little shorter than optimal, rather than a little longer.