Driggs, Idaho is a 2 1/2 hour drive from Big Sky if the weather is cooperating. The drive is through some beautiful country that is very sparsely populated. If you are driving late at night and end up in a snow bank it might be a while until someone else passes and you won't be able to use your cell phone. Much of the drive is wind prone and you can experience zero visibility even if its not snowing. Plan on taking 5 to 6 hours if necessary.
Grand Targhee gets more good powder than any ski area I know of. Its biggest draw back is its terrain is moderate but for people wanting to learn how to ski powder this is a big plus.
Your other options from Big Sky are Moonlight Basin (which is right next to Big Sky), Bridger Bowl and maybe Discovery Basin. I would rule out Discovery for your family. Bridger Bowl if a fun hill but it lacks advance terrain. Basically, everything there is either intermediate or expert. Moonlight has a great deal of intermediate terrain and hopefully some true advance & expert terrain next year.
Driggs is 145 miles from Big Sky (185 from Bozeman). If the winds aren't blowing and its not snowing you can do 65-75 MPH most of the way. 3 hours is probably a saner estimate of the time it would take in ideal winter conditions for someone not familiar with the roads and 4 hours if you are stopping along the way (though there aren't many places to stop especially in Idaho on a Sunday). Because of the weather you might encounter you need a to allow for a 5 to 6 hour drive. It took me 6 to get to Driggs from Bozeman when I went to the gathering and I made it from Driggs to Bozeman in 3 on the way back.