I will agree with every sentiment echoed so far. I currently am a volunteer ski patroller (since 2001 at various resorts), I have lived and worked in ski towns, and I grew up in California and learned to ski in Lake Tahoe. It is expensive. However, my wife and I now backcountry ski quite a bit and I ski patrol which gets us free season passes. I would encourage you and your family to explore the snow, and then determine if you want to start skiing or snowboarding.
For a cheap introduction, look into nordic skiing and/or snowshoeing. ANYONE can showshoe. It is glorified hiking on snow. However, you can explore areas in the winter and see some beautiful sights. Rent snowshoes (~$10 for the day) and try it out with the family. Maybe even splurge for a guided snowshoe trip? Introduce your family to the snow slowly. Also, with kids, do it when the weather is nice. If it is going to be a blizzard and cold, most kids won't want to learn in those conditions. Go out this spring if you are there and explore while it is sunny. Then, if the family likes being in the snow, consider nordic skiing or going to a ski resort. Nordic skiing is a ton of fun. My wife and I picked up setups (skis w/ bindings, poles, boots) while living in Michigan for less than $100 for each of us....we still use them 6 years later. We don't really go to nordic ski areas, we just go out and explore our local National Forests and public land, which is FREE! There is an amazing amount of public access around Lake Tahoe. If you have nordic skis (or snowshoes), common sense, outdoorsy skills (map reading, compass, etc.), and your family likes to explore, nordic skiing can be a great way to get out. And, if you have a dog, they can usually come along for some exercise. If you go this route, look for traditional nordic ski setups (not skate skiing) and look for packages or ski setups labelled as "touring" or "backcountry". These are nordic skis that usually have a little wider ski and possible metal edges for more cross country travel. You should be able to hit up ski swaps and get your entire family skis with bindings, poles, and boots for <$100 each. After that, no lift tickets, no extra expenses, just fun in the outdoors in winter.
Okay, so now maybe your family likes snow and wants to try downhill skiing. Bite the bullet and pay for lessons. I learned to ski at Alpine Meadows ski area in Lake Tahoe when I was 8. Cub Scout trips once a year. Like others have said, look for package deals of rentals, lessons, lift tickets. Try it a few times. If your kids (and you and your spouse) like it, consider buying gear at a ski swap or look for sales in the spring and summer when shops are selling their demo gear from the previous year. Also, look for shops that do season long rentals or demos for the kids (because as your kids grow they will need new gear). Our family in Michigan has a local shop that does a one time, up front payment of $400 for skis, boots, and poles for each child. But, after that, they only pay $100 for each year and the children get new demo equipment in the proper size as they grow. They get to keep the gear for the entire season and they turn it in at the end and the shop sells it off as they replace their demo fleet. There is also an option to buy the equipment at the end of the season.
Then, if your kids and you really start to enjoy skiing and/or snowboarding, then you will be hooked and you will sell other parts of your life to get season passes, you might want to volunteer as a ski patroller to get free passes, you will join websites focused on vehicle travel to modify your vehicle and buy a camper for said vehicle where you, your wife, and family can travel to ski areas and sleep in their parking lots and ski as much as possible (oh wait, I digress). In all honesty, introduce winter activity slowly and then build from there -- it is a lifetime activity.