K.L.
Hi, I'm sorry but I don't know much about Atsugi, per say, but here is the weather for Atsugi, so perhaps this may help you decide what the pack?
http://www.wunderground.com/global/stations/47679.html
I used to live in Hiroshima, Japan, before kids. (I have two kids and two dogs as well.) From my understanding, most people rent condos or apartments and not homes in Hiroshima. It also depends on how "urban" the particular area is in Atsugi.
I looked on line and they have a Naval Base there so perhaps you'll find more 'westernized' things there? When do you leave? One thing I'd recommend bringing is WARM CLOTHES! If you are not the size of a typical Japanese person, then I'd recommend bringing warm clothes for yourself because it was hard to find "western sizes" in clothes in Hiroshima.
Why are you moving there? Company? Job? Navy? If so, perhaps there's someone or some resource from there that you can access? Also, shoes are hard to find in larger sizes. They did have a GAP and H&M in Hiroshima. They also had a store called Uniqlo that was like Old Navy that carried larger sizes in warm clothes.
I used to live in an apartment, but it was 3 bedrooms (3LDK as it was called; "living, dining and kitchen" -- 3 was the number of bedrooms.) SUPER small if you ask me, maybe about 1,000 square feet but for Japanese standards, it was big. It was more like a "mini house" but it was in a high rise apartment building. Finding somewhere that's suitable for your dogs as well might pose a challenge. It also depends on how big your dogs are.
Try this website:
http://www.repropertymanagement.com/JP/x/Atsugi.aspx
or this one:
http://www.e-houses-for-rent.com/JP/x/Atsugi.aspx
How old are your kids? Will they be going to school there as well? Japan can be a very exciting place, but it may help to buy some books on "common kanji" or "intro to speaking Japanese" as well. Most big cities have some people that speak English but as you get further in the rural areas, it gets harder to find English speaking people.
Best of luck to you and your family.