The Baby Surprise Jacket is unusual in that the instructions are the same no matter what type of yarn you use - same stitch numbers, etc. The way to get different sizes is to use thicker or thinner yarn.
I'd never recommend using a different sized needle to get a certain tension with the yarn you're using. The best and correct tension for a yarn is the one recommended by the manufacturer. So the best idea is to knit a swatch in stockinette with your chosen yarn, make sure your tension is as recommended on the ball band, and then knit the jacket with those needles. Change yarn for a different size in this jacket, not needles.
I've recently knitted a Baby Surprise Jacket using Plymouth Encore Colorspun and mine is about 6-9 month size. Another one, knitted with a DK wool (tension 22sts to 10cm), is about 3-6 month size, and I've also knitted one in Peter Pan Softblend DK baby yarn, which is newborn size. The recommended tension, over stockinette stitch, for the Peter Pan is 24sts to 10cm (4 inches).
So for a newborn size, a yarn that knits to 24sts tension would probably be the one to go for. A soft yarn is best, as this jacket, being all garter stitch, is quite thick.
I bought my pattern from Schoolhouse Press and also googled "Baby Surprise notes" to find some very helpful notes by Dawn Adcock, which you use in conjunction with your pattern. The pattern itself is more like a few jottings, not a regular pattern, and those notes are invaluable.
For a newborn, I think a slightly shorter jacket is better. You can find info on this modification and others - such as lengthening the sleeves - by doing a Google search.
I also joined a yahoo group called Knittingbabysurprise - lots of help there, and files with notes from previous knitalongs. There are also many photos of Baby Surprise jackets on flickr.com.