Forget South Lake. In the winter, even with good weather a theoretical "40 minute drive" will easily be 3 hours (no fun). Come to North Tahoe, you will vastly open up your options. You will have more mountains to choose from, better housing options, more places to eat, it will be less expensive, be closer to everything, the list goes on and on. Just do it.
There is also a lot of good advice above, so listen to it.
Now what I would like to add is DO NOT bring your family to Northstar. Let me explain.
Northstar does have a great kids / ski school program, They also have the best village, best grooming, best nightlife, best deals (stay and ride free, lots of discounts), most convenient housing, most convenient dining, nicest amenities, best tree skiing, the list goes on and on. BUT it is by far the most crowded and because of the massive numbers of people (especially in beginner zones) is flat out dangerous. I would never bring a brand new skier to Northstar.
A good alternative is Tahoe Donner. Tahoe Donner is a hokey tiny ski hill (2 lifts) in Truckee, but their instruction is top notch, kids rental equipment is the best in the area, never is crowded and tickets can be dirt cheap ($5). It's worth giving it a go a least for your first day. Everyone will be happier because of it. It is small and boring but if you are serious about your kids learning to ski, it's the best choice.
Squaw is also a good choice, but is very expensive. There is also a very unique vibe at Squaw that you will either love or hate.
One more thing... AWD / 4x4 is useless in the snow without really good all season tires or dedicated snows. Considering you are renting a SUV in AZ, you will get neither. Don't be surprised if you have to chain up your 4x4 SUV... I've helped people chain up their Subaru's (sad yes) because of inadequate tires.