Probably the first question that's going to pop to mind when it comes to kids ski gear is whether you should buy or rent. I strongly believe you should buy your own gear. Where available, you can also rent from shops that let you keep the gear all season (you trade up at the end of the year), though this may not actually be cheaper.
The worst choice is to rent day-by-day at the mountain. Forget the fact you'll eventually pay more. What really matters is (1) you'll be exhausting your child's precious and limited good-to-go time in long, slow rental lines, and (2) you won't have the gear available at home for your kids to play with. Don't underestimate the impact of either of these points.
The advantages of having your own gear at home, in particular, are huge. Not only do you get to familiarize your child with their skis and boots (which become their skis and boots), you also get the joy of watching them ask to put on their boots and skis and tromp around your living room. Without ever touching snow, your kids can pick up a lot of ski skills right at home, including side-stepping, making circles (one leg follows the other), and you pushing them across the floor.
Be careful with that last one, as your kids are likely to love it so much they'll have you on hands and knees for the rest of the winter...
skis & boots
Anyhow, skiing at home is a subject we'll revisit later. For now, let's return to gear. For ages 0-24 months (depending on foot size), forget about rigid boots and bindings and get a pair of Karhu Kabooms (try ebay) or Lucky Bums Beginner Skis. The 85cm Kabooms are really nice but hard to find. The Lucky Bums are similar but shorter.
Both employ a 'universal' binding that will accept any snow boot. Realistically, at this age not much actual skiing is going to take place, but if you want to hold your child upright on a green run to let them feel the whoosh of skiing (good sound effects are mandatory), these are a great choice. They're also excellent for skiing at home, as your child will be able to step into the bindings with shoes or without, without any help from you, which is kind of invaluable.
Right around 24 months your child's foot will get big enough to fit inside the smallest hard-shell ski boots (I like technica's rear-entry Tecnica Easy T), allowing you to upgrade to an Alpine ski with a step-in binding. The boots will probably still be slightly large, upping the terror factor on the chairlift, so use thick kids socks. On the upside, they'll probably fit next year too.
Be aware, at these early ages, that rear-entry is practically a mandatory ski boot feature. Try stuffing toddler feet into a boot like Tecnica's RJ Super 2 (a good choice later), and you'll see what I mean immediately.
This pair of skis and boots will be your kids' first 'real' pair, and should be good for when they're 3 as well (when skiing off a leash becomes possible). Ski length should probably be no shorter than 80cm. Browsing kids' skis online reveals a dizzying array of choices. I like Dynastar's Team Speed. Avoid junior versions of adult skis—they're needlessly expensive at this age. Search around for deals that include an appropriate binding, which should go down to a DIN 0.5.
Somewhere between ages 4 and 5, with regular snow time, a child can progress to become a credible little skier, able to control their speed with turns on a variety of pitches, with a natural body position. More advanced skiers in this age range may flirt with parallel turns, moving away from basing everything on the wedge. This, incidentally, is the age at which most ski school programs will start taking kids, and it's perfectly reasonable to wait until this age before you start putting your own child on snow.
Specifically for those who want to start earlier, however, we have a few accessories which will prove to be essential. Chief among them is the Edgie Wedgie. Get at least two of these right now, and keep a backup in your pocket at all times! Children below about 4 do not have the strength/coordination to manage independent skis. The Edgie Wedgie connects the ski tips, doing the bulk of the work in creating and maintaining a wedge shape for your child. For the youngest skiers, be sure to tie a knot in the rubber tubing to shorten its length.
You will also likely want a leash and a harness. The harness is an especially important piece of gear, as it allows you to hold onto your child while skiing without crouching excessively. Your back will thank you! Get Lucky Bums' Harness, or look for a harness integrated into a vest. Note that harnesses of this type are not equivalent in security to climbing harnesses—don't count on them in life-or-death situations, such as on a chairlift.
In addition to giving you an easy way to grab your kid, you can also use your kid's harness to attach a leash to them. Any kind of leash will do; a dog leash, or six to eight feet of nylon webbing. Some parents like to use dual-leash systems, which allow you to turn the child's upper body and thus control their direction by 'steering' them. I'd stick with a single leash. Dual leashes are more complicated, and they only serve to worsen the liabilities of putting your child on a leash, which is that your child starts to depend on it...
helmet, & goggles
As of 2012 I still haven't seen compelling evidence that wearing a helmet is significantly safer for skiers like me. That said, I have no hesitation in recommending you put your child in a helmet as soon as you can find one that fits (this will likely be right around 24 months old, and certainly by age 3). Helmets are nice and warm, they offer great wind and weather protection, they hold goggles in place on tiny faces, and they absolutely reduce crying when your child falls. These perks are big enough on their own to justify a helmet even if helmets didn't offer any safety benefits whatsoever (and I doubt that's the case). Giro's Slingshot is a good kids' model; start with an extra small.
Note that if you choose to not wear a helmet yourself, at some point your clever child is going to notice it, and ask why you forgot your helmet. I leave that conversation to your own creativity. As for goggles, I recommend them, rather than sunglasses, because they offer wind protection. Smith makes a number of fine choices in their Junior series. Do keep a pair of sunglasses in your kid's coat pocket, however, for when they eat lunch, or for walking around the base of the mountain. It's a good habit to get them used to eye protection at all times in the mountains.
Jackets don't need hoods if you're using a helmet, and you won't need a ski hat either. After age 3 or so kids are big enough to fit either full-bib (overall) or regular ski pants. Both have their advantages. Below age three, you'll want to get full-bib pants. Note that bibs prevent snow from getting to your kid's sensitive torso when they fall, and this can be very important. If you have a younger sibling waiting in the wings, there is some peace of mind in knowing that clothing and gear will be handed down for later use. If not, you can sometimes squeeze an extra year out of clothing by buying slightly large. Regardless, size the jacket correctly. Too-long sleeves will drive you insane when it comes to getting gloves to stay on.
Whatever clothing you settle on, your kid will likely become attached to it, and expect to wear exactly the same outfit regardless of weather. This is good, in the sense that it adds to the routine and the familiarity. But it can bite you if your child ends up woefully over or under-dressed. So give some thought to the expected conditions you'll see the most. Here in Southern California, for example, it's almost never truly cold, even in the mountains. In any case, stick with the same outerwear day after day, and vary under layers as needed to deal with temperatures. Be sure you have a least one zip-neck shirt.
gloves & socks
Gloves and/or mittens are a source of endless frustration when it comes to skiing with kids. The problem is they're hard (or just impossible) to get on correctly, and they constantly fall off. I don't have any great solutions to this problem. Start with mittens and stick with them until at least age 4. Give L-Bow Gloves a try. They're far from perfect, but they do keep snow away from your kid's wrists. If weather and your child are okay with it, fleece mitts are easier to use than heavy waterproof mittens. Fleece of course is not waterproof, but fleece mitts are actually functional, rather than just blobs stuck to the ends of your child's arms.
Kids ski socks, in my experience, are almost universally too thick. This is good news when you're putting your 2-year-old on the snow, as the socks help pad out ski boots that would otherwise be too big. But once your child gets a little older, thick socks equals foot constriction and pain, which must be avoided at all costs. I just use my kid's cotton street socks, which are sheer and thin. This works even in cold weather, as you'll be exposing young kids to bitter cold (if at all) only in the smallest of doses. In that regard, always make sure kids' ski boots are pre-warmed before you stick tiny toes into them...
Sunscreen is an unavoidable part of skiing, I tell my son. No one likes it, but we have to do it. I worry incessantly about chemical exposures, so I do try to minimize the amount I put on my son's face, and I try to keep it away from his mouth, but at some point you just have to shrug and cross your fingers. There are websites that rate sunscreens based on their effectiveness and chemical content. It's not clear how accurate those ratings actually are, though it is probably true that some sunscreens are better than others where kids are concerned.
If you're curious, I use Coppertone's sport lotion, SPF 30, for both of us. Since my son is always skiing in a helmet and goggles, I only put lotion on his nose and his cheeks. I also try to sit him facing away from the sun at lunch time, or else we eat indoors. Don't forget about lip protection. I put a Blistex in my son's ski pants pocket. Nanoscale metals (ie, micronized Titanium) are an extra concern in lip products, but unfortunately almost unavoidable. You can use Vaseline Lip Therapy or Burt's Bees Honey for the least chemical exposure worries, but these don't offer any sun protection.
your own gear choices
Leave the ski poles at home. I've tried going with and without, and having your hands free at all times outweighs the negatives. Your child, of course, won't need ski poles for a while—not until parallel turns start appearing. And even then, you'll probably use ski poles intermittently, only on days/runs that are more technique-focused. Bring a backpack small enough to wear comfortably on a chairlift, but capable of carrying your kid's skis and an extra pair of shoes (for them). Strapping skis to the outside of a tiny pack works just as well as putting them inside a larger pack.
I find it works better to use skinny skis rather than your fattest powder boards. This is especially true on Magic Carpet days, but generally true elsewhere as well: narrower skis are lighter and more nimble. I'm always warmer than expected when I ski with my kids, so dress a little lighter than normal. Especially when you're just starting out, you'll find yourself expending a great deal of energy, and you'll be at the base of the mountain, where it's warmest. In both clothing and equipment, stay as light and as simple as you can. Overloading yourself only makes things more difficult.
If there is a great deal of walking to be done to get to the chairlift (and this is something that should factor heavily into your choice of where to go), consider bringing an Ergo child carrier. These are my favorite, and they will fit kids up to almost 3 years old, more or less. They are structurally solid but completely structureless, so you can roll it up and stash it in your pack when not needed. Remember: you will be carrying your two year old one way or another.
organization & final thoughts
Getting yourself, your child, and all the necessary gear to the ski hill is a daunting challenge. Not to mention the effort required in finding and buying everything to begin with. Be sure to devise a reliable system that helps prevent you from forgetting things. This may mean stocking a kids-specific ski bag, or even buying backups for those inevitable times when you do forget (or lose) something important.
Finally, if you've gotten this far, congratulations! You've already done a huge part of the work. We'll talk a little more about gear, going over things individually in more detail in later articles, but for now, you're just about ready to get your kid out on snow...
Of Mountains and Molehills
First Steps: Kids on Snow
Finding Appropriate Terrain
Skiing With Kids: Dry Land Training