Definitely separate. Definitely. You'll have a bunch of guys there packing and there is no way to watch their every move so it needs to be utterly clear what they should and should not pack.
It's a pain, but we moved things so that everything in one small storage space was Express (UAB), 2 bedrooms, the living room and kitchen where regular move (HHG), and everything else was storage. Then we shut and actually taped over the doors for the rooms that each set of movers wasn't supposed to do. Out UAB was first, then HHG, then storage. We did leave a few major pieces of furniture in the "wrong" rooms but those would have been easy to notice being carried out on the wrong day.
Try to split up items so you have everything you need to get by in both your UAB and your HHG. It is not unheard of for the HHG to arrive first, so if you put every pot and pan and towel in your UAB, you'd then be screwed. Hedge your bets and make sure both have what you need to live.
Try to schedule your UAB well ahead of when you leave. It's easier to live without those things on that end, rather than this end.
Do not even consider bringing a car. EOD. Just don't. So many reasons. And cars here are cheap. A very decent car can be had for $2000 and something that will cause you no problems and last your entire tour just fine, but not look so pretty, can be found for $1000 or less. I don't recall registration being as pricey as helo wifey mentions, but a lot of that depends on how long is left on the various weird insurance things they require, as well as the size of the car (smaller car=cheaper registration). We bought our cars at a dealership and they handled some of the convoluted registration process for us, and I think we paid about $800 on each car in addition to the sticker price.
For clothes, there are so many places online that ship that make it easy, so personally, I wouldn't worry too much about that. Fit can be an issue there though, admitedly. You'll probably also fly back to the States at least once during your tour so you can always stock up then.
Opinions vary, but I say you should bring as much luggage as you can stand to haul (and don't have to pay for, though I'm told the Navy will reimburse if you exceed the airlines limits). Most airlines have generous allowances for luggage if traveling on orders. I think we were on Delta and the AD member was allowed 3 70 pound bags and I was allowed 70 pound bags, both of which are more than the normal person gets without being charged. We brought all 5. Not being familiar with weather (and maybe arriving during a seasonal change and therefore needing items to cover different conditions), not knowing what squadron events might be held before the rest of our stuff arrived, etc., having to have a zillion uniform items just in case Husband needed them before stuff arrive, etc., were all reasons to just bring as much as possible.
On that note, keep in mind you will almost certainly be living in a hotel room for at least a couple weeks. So luggage isn't just for clothes. Bring toys for the kids (you have kids, I think?) and things that will make that suck a little less for you. For example, we packed out wireless router since the Lodge had internet but not Wifi. Convenience items take some of the sting out of living in a hotel for a month, so toss them into a suitcase. Even something like having your own pillow might be nice if you are a picky sleeper. Keep that type of stuff in mind when packing your luggage.
Do you have thoughts about living on vs. off base? If you want to live on, you might see if your sponsor can get you in touch with someone else who does. In Atsugi at least, the floors are linoleum so people often recommend bring rugs, but it's nice to know sizes, if possible. Also, I hear people on base talking about needing to buy curtains and recommending people arrive with them if possible, so gouge like that is nice to have.
If you want to live off base, have money in savings. There are all sorts of deposits and fees--way more than stateside rentals. Some are reimbursable, and sometimes you can even get the Navy to pay some up front so you are never out that money even temporarily, but some will be out of pocket and for the rest, it is much easier to just pay it and get it back then try to deal with a landlord and Housing and PSD.
Also, if you are considering offbase, see if someone can go to housing and get a list of current available places for you. I won't help you actually pick a place, but it will give you a great idea of how much sqft. your OHA will buy for you.
We did take the opportunity to purge a lot of household crap, but I erred on the side of keeping things. As long as you aren't over the weight allowance, it's free so it doesn't really hurt anything to have something sit in storage and then get thrown away when you get to your next house. I was hesitant to toss something and then end up having to buy a new one in 3 years when I discovered that the chair I got rid of would fit perfectly in X location.