My son started to ski when he was 2 - and just barely 2 as his birthday is in December. We got him a pair of Trek (or was it Trak?) skis with fishscale bottoms and no edges and NO POLES. His first on-ski experience was on the living room carpet on Christmas morning and his second was walking around on snow in our flat backyard that afternoon. The walking around on the flat is extremely important and worth the patience that it will take while he gets used to the skis. The fishscale bottoms were a huge plus. I think one of the most frustrating things for little kids is slipping backwards.
Your characterization of "playing" will also be a huge plus. I consider "adult expectations" to be the single greatest impediment to small children enjoying learning to ski. The initial objective for your son will be to move across the snow to where you are, at which point you will either catch him and give him a big hug or laugh with him when he falls onto the snow before reaching you. If you expect him to control his speed, turn, stop, press forward in his boots, distribute his weight evenly from side to side and front to back, keep his hands up and in front of him, or any of the rest of that junk, you will be disappointed, he will sense it, and he will not have fun. And he MIGHT get turned off to skiing.
Everybody who's had success introducing their 2 or 3-year olds to skiing will have a different methodology that worked. Many will insist that theirs is the best or the only way. It isn't. What will work is what will work for YOUR CHILD. And the only sure-fire way to assess whether it has worked, at that age, is when he tells you he wants to go skiing again. Don't fall into the sometimes convincing trap that some will set when they tell you that you must teach him skills that he can build upon later so that he will avoid bad habits and ski "correctly". That's nonsense.
In our case, I skied with my son between my legs, holding him under his arms(for what seemed like a hundred miles), until I felt that I could lower his weight down onto his skis without him collapsing into a giggling mass. When he felt stable I would let him move out away from me, ski down ahead of him, turn around and let him ski into me. About half the time, in the beginning, he would tip over before he got to me. No big deal. Sometimes he would be eager to get right up and try it again, other times he would lie there eating snow or making angels. It was all part of the experience.
The next year, at 3, he could ski down from the top at our little local area, dragging a stick and making rooster tails in the snow. He never really learned to turn, he just followed the trail. Sometimes I would ski beside him holding the lower edge of the back of his jacket to help him control his speed. (This was more for my peace of mind than for his.) Many people have success with harnesses. Some people rant against them. It is true that small kids can get tangled in the straps, but that's usually because of the adult's ineptitude and clumsiness.
The next year, at 4, he skied from the top of Mount Mansfield in Stowe, VT., down the 4 mile long Toll Road trail. it took about an hour and a half and he never really made it to the bottom. He was so exhausted he literally fell asleep and skied off the trail into a snowbank about 100 yards from the bottom.
And so it went. When it came time for him to actually "learn" stuff, at age 5, even though my wife and I are both instructors, we signed him up for the kiddie classes at the local area. WHEN THEY'RE READY, kids have a great time with others their age, especially if the ski school has the same attitude that you do - that it is "playing".
Have a ball. My son is almost 19 now, he instructed after school kids programs all through high school, got his Level 1 certification, and is an incredible all-mountain skier and a blast to ski with. But some of my fondest memories are of skiing with him, at his pace(very important), when he was 4, 5, and 6. Do what works for your boy and it will work for you, too.