welcome to Epicski!
You might want to try the 'Beginner Zone' forum as well: http://www.epicski.com/f/56/beginner-zone , and the 'Resorts, Conditions, and Travel' forum for travel-specific advice: http://www.epicski.com/f/21/resorts-conditions-travel
This is just a bunch of general things off the top of my head.
I'm not sure how long you are looking at going. If it's just a few days, it's probably way more cost-effective to rent gear. Rental ski boots are never the best, but most people manage to live through the experience. :-)
If you're planning a longer trip, and you think you'll go again in the future, I'd give some serious consideration to buying your own ski boots. They are generally much more comfortable and better-fitting, and you can keep them for many years (though your kids may grow out of theirs). Rental skis are just fine for beginners, and dragging them on a plane is annoying. If there isn't a good bootfitting shop in the Atlanta area, there are plenty of them around any major Western ski resort, and they could set you up in a few hours, especially with an advance appointment. I think if you go to the 'ask a boot guy' forum, there is a list of registered shops there.
If you're renting skis and other gear, it may be a lot cheaper to get it somewhere that isn't right at the mountain. There are almost always ski shops in a nearby town with better pricing. However, some ski areas have aggressively-priced packages for tickets, lessons, and rentals all together, so you should look at those as well.
You'll need some winter clothing. The trick to staying warm is layering. Most people start with some kind of long underwear 'base' layer shirt and pants, then you can add things like a fleece or down vest (or long-sleeved fleece top or sweater), and then an outer 'shell' jacket and ski pants that repel wind and water. Add more layers as needed. AVOID ANYTHING MADE OF COTTON. You want wool, down, or polyester fleece (or other synthetic fibers). If you have a cotton layer on and it gets wet or you start sweating, you'll stay wet and cold and miserable all day. Wool and synthetic layers will 'wick' moisture away and actually dry out while you are wearing them.
For 'other' clothing: you need insulated, waterproof gloves. Also eye protection -- you can get away with sunglasses if it's warm, but if it's cold you will probably want dedicated ski goggles. You'll also need something on your head and to cover your ears, either a warm hat or a ski helmet. (You can rent helmets at most resorts; your kids may be required to wear them in lessons.) You may want scarves or neck gaiters, or a face mask/balaclava. You'll also want wool or synthetic ski socks -- again, NOT COTTON, no matter how comfy they feel when they're warm and dry.
Where to go:
Right now the snow conditions are pretty grim almost everywhere out West. So this is just general info not taking into account where conditions are best right this second. If you have some flexibility in your plans, you might want to wait another month or two and see where they're getting snow. If you're looking for a Christmas->New Years trip, it's not looking real good at most of the big resorts, but that may change quickly. If you're willing to tolerate last-minute booking craziness, flying to wherever it's snowing (or about to snow) often leads to good results.
There are lots of other places you could go, but these are the biggest destination resort areas in the US.
Utah generally gets the most snow, and the big resorts there have lots of 'extreme' terrain (which, of course, you won't be using as a beginner). The resorts near SLC (in the 'cottonwood canyons' area -- Snowbird, Alta, Brighton, Solitude) are pretty bare-bones in terms of facilities, but there are lots of nice rental condos and hotels in SLC and the surrounding suburbs. Park City is a cool little town (the Sundance film festival is there) with several ski areas nearby. Park City Mountain Resort is accessible directly from downtown. The Canyons is huge. Deer Valley is very upscale, if you're into that sort of thing. Again, lots of condos around Park City.
Colorado has several major ski 'regions'. 'Summit County' is the area west of Denver that has a number of well-known ski areas (such as Vail, Beaver Creek, Breckenridge, and Arapahoe Basin) and several smaller ones (Keystone, Copper, Loveland). Aspen is further west, and is fairly isolated, but Aspen itself has four mountains, one of which is pretty much a dedicated beginner's area, and all of which offer terrific skiing. (Not cheap, though.) South of Denver there are some smaller (by Western standards) areas like Telluride, Durango, and Wolf Creek.
Lake Tahoe is another popular destination. Great views, usually cheap flights to Reno, lots of relatively cheap accommodations, lots of different ski areas nearby. It's roughly split into the 'north shore' (Squaw, Northstar, Alpine Meadows, Mount Rose) and the 'south shore' (Heavenly, Kirkwood).
There are a few other major areas (Jackson Hole, WY; Sun Valley, Idaho; Mammoth, CA; Mount Hood and Mount Baker in the PNW; Whistler, Canada where the 2010 Olympics were) that are somewhat less mainstream. These are generally a bit more isolated (or, in the case of Whistler, in another country). Jackson may also not be the best choice for beginners, as it is best known for its crazy super-expert terrain. There is some nice terrain up here in New England, too. But if you have to fly anyway, IMO you should go out West. The lift tickets aren't that much cheaper.
One other caveat: altitude. Even if you're in decent shape, it will take at least a couple days to feel normal doing athletic activity on a big mountain. Some people get altitude sickness pretty bad, especially if you are sleeping much above 5000-6000 feet. This was a bit of an issue for me in Summit County, CO, but not so much in Tahoe or Utah (where I was staying at a much lower elevation than the ski resorts were at). Usually it goes away within a day or two, but it's not much fun.
If you pick a resort and stay "slopeside" (or even 'ski in/ski out'), you won't have to waste time driving around and parking. But it's also usually the most expensive. Generally you trade off niceness and convenience of where you're staying versus price. Being willing to drive 20 minutes each way may cut prices in half (or more). A condo with a kitchen will give you more space and let you go to the store and cook your own meals, if you're looking to save a few bucks. Going at off-peak times will almost always be cheaper than during the holidays. When I go somewhere I'm basically just looking for a clean bed to sleep in between ski days, but there's usually a huge variety of places to stay around any large resort.
All of the areas I mentioned above are within a few hours drive of a major airport. (Aspen is probably the furthest unless you fly into their little local airport directly, which usually costs a fortune.) Depending on exactly where you go, you may be able to get away without a rental car, especially if you're staying right at one specific resort. e.g. if you stay at Vail, you can get a shuttle from the airport to there, and then just take the free buses around Vail Village or over to Beaver Creek and back. If you want to travel around at all, a car makes things much more convenient, and you don't have to deal with airport shuttles, buses, etc. If you do rent a vehicle and you're going to be driving around in the mountains, I recommend something with 4WD/AWD. Take it real slow if you have to drive in inclement weather, especially if you're not experienced in those conditions.
I have not generally found 'ski and stay' packages from the resorts themselves (where they bundle accommodations with lift tickets) to be competitively priced, but it can't hurt to check them out. For flights and hotels I usually use online search engines like hotels.com, Orbitz, Kayak, etc. (Keep in mind that Southwest doesn't list their flights on any of those.)
You will need lift tickets unless you want to walk up the mountain. :-)
Try Googling or searching around Epicski for deals on different mountains. (Try things like "<mountain> discount lift tickets".) For longer trips, sometimes multi-day packages direct from the resort can be pretty reasonable. Other times you can do much better buying discount tickets online or from local sources off the mountain.
I'd strongly recommend lessons if you're just starting out. Many resorts offer packages with beginner lessons, equipment rentals, and lift tickets bundled together at a discount price. Those would normally be class/group lessons, where you're in groups with other guests. A pricier option is private or semi-private lessons. As a beginner I'd recommend a few days of group lessons, and then think about private lessons if you want more. But private lessons are great if you have the money for them.
If you stay in an area with many resorts nearby (such as Park City, UT or the Summit County area of CO), you may want to consider visiting several different ski areas. During a longer trip you may also want to schedule in some days off to just see the area and get some recovery time in!