I'm a fairly strict Orthodox Jew, so if you have specific questions, feel free to PM me.
Circumcision care -- usually the mohel will leave very precise directions. Generally, all that's involved is spreading some antibiotic ointment and covering with a guaze pad for two or three days. Most of the foreskin is retracted, but not removed, in a Jewish circumcision, so while the penile head will be red and swollen, there isn't the same amount of "wound tissue" as in a traditional American circumcision.
Re: the contact rules: one thing most people have trouble getting used to is that not only won't the couple touch each other, they won't pass things to each other either. That can make for some awkward situations with a newborn, and I'm sure they'd appreciate it if you were on the ball about taking the baby from one and handing him to the other when indicated.
When it comes to the Sabbath, they'll probably ask you to do all the "work" (very technical religious term; encompasses most activities involved in farming, cooking, etc., but what you're more likely to be affected by is electricity, hot water) involved in baby care, but there's a fine line -- some people won't ASK you to do anything for their own comfort but might appreciate it if you did. It can get very sensitive if you're dealing with ultra-Orthodox Jews -- the post-partum mother can benefit directly from something you do for her even if (in some cases) she can't ask you to do it, but it's more complicated for someone who is not somehow medically impaired. So if you see the father trying to read in a dim room, he can't ask you to turn on the light -- and it could be that he wouldn't even be able to use the light if you did turn it on unless you "used" it first -- looked for something on the bookcase, or something like that. Best practice would be to talk about it candidly before hand. Sabbath starts just before sundown on Friday.
Cultural dos and don'ts:
Don't hug the father, or pat him on the back, or initiate any physical contact (outside of things like handing him the baby).
They would probably appreciate it if you dressed "modestly." For very Orthodox people, that means sleeves below the elbow, skirts below the knee (though they'd probably be fine with not-too-tight pants), and necklines at or near the collarbone. Don't know how strict this couple is, though; you can definitely take your cues from how you've seen the mother dress.
Ask before doing anything in the kitchen. The kashrut rules are very complex and you might not realize you were doing anything wrong before rendering an expensive appliance useless to them.
Don't set anything down on top of a book with Hebrew lettering (only a problem really for religious books, but better safe than sorry).
That's all I can think of off the top of my head.
