I went with dd1 (age 10) in Feb. It was her first time, and I haven't been since childhood.
wrt to the ski gear, cheapy places like Lidl and Tchibo do ski stuff early in the year (Nov or so) which is worth keeping an eye out for. Most of dd1's gear came from Tchibo and wasn't brill quality, but perfectly okay. I borrowed mine from a friend. You need thermals (several pairs if you can round them up, as they do get a bit sweaty), salopettes or trousers, a fleece or two and a ski jacket. Also winter-quality walking boots, ski gloves, ski socks, hat, neck thing, goggles or serious sunglasses. Don't be shy about asking to borrow stuff -- most people don't ski for more than a week per year, so there's a lot of gear lying unused in cupboards the rest of the time! The skis, skiboots and helmet (if needed) can be hired.
We went to Austria which is meant to be friendlier than France, tho' we went with Mark Warner so hardly had to talk to the natives. For your first time I'd recommend going for something like MW (lots of people were complaining about Esprit on another thread, so maybe not with them) because it's all so easy everything done for you. The Austrians are pretty friendly, tho' the ski instructors are all different nationalities. If you do go, don't make the mistake I did and book St Anton it's a fab resort but has no green runs at all, and most of their blue runs would be red anywhere else! Tho if you're gung-ho it can work out -- dd1 was going down blue runs on her second day on skis!
I don't think joint classes for you and the kids will work, tbh. I had a couple of dry-ski lessons beforehand with dd1, but she's much older than your kids. I'd do a couple of lessons for you and dh before you go, just to get you into the swing of it and learn to snowplough etc. and then book the kids into ski-school or childcare.
Def go, it is the most fantastic fun!