I understand the desire to keep all of your sets intact during the move, but keep in mind that depending on how long you take to pack the sets and keep them protected, it might be a better use of time to simply repair them when you reach the new place.
My suggestion for getting them moved is similar to some other folks, especially concerning your big sets. Put your set in a bag, tie it up, dump it in a box, then put padding (newspaper, bubble wrap, etc) around the in-the-bag set. Your set is secure enough for the move now, and if any pieces fall off (which they will), they're not only in the bag with the rest of the set, but you also don't run the chance of throwing any pieces away with your packing material, which could easily happen if you put your padding inside the plastic bag with the set.
For the big sets, I'd probably throw a little bit of padding in the box underneath the set too to cut down on car vibration/bumps. I wouldn't worry about the little sets as much, they usually hold together well and they're easier to put back together. I'd still bag them individually, but maybe do a layer of small sets in a box, then a layer of padding, then another layer of sets, etc.
One last thing- it might not be useful advice to members of this forum since we all have mountains of shipping materials lying around, but t-shirts are awesome for packing up, especially for fragile stuff like plates or glasses; might work for some of the Lego sets too. There's also the added advantage that once all your breakables are ready to go, half of your clothes have been packed too.