I'm constantly being asked about what to bring/not bring with you when you relocate to Kwaj. This is a tricky question. First of all, it's really a personal preference choice and second, it depends on what company your hire on with and what your status is.
If you are a KRS employee, you can bring 750 pounds unnacompanied and 1500 if you are accompanied. I think you can add X amount for each child as well but it's not a large number.
USAKA and AAFES employees get something outrageous like 15,000 pounds and a house no matter what your status. In this case, you can bring everything if you want.
I've seen people come out here with two checked bags from the plane and that's it.
The items I would definately suggest you bring would be electronics. Computers, TV's, etc. These are not readily available here. You may get lucky and get something good when someone PCS's but the store does not carry much if any selection.
The one thing I do recommend is to mail bedding and towels to yourself or your sponsor before you come so they are here when you get here. (At least two weeks) Again, the selection is very limited here and the hospitality kit includes these items but you only get to borrow them for a limited time and they are not exactly nice.
As far as bikes go...I would almost suggest you wait until you get here. It could take almost two months for your stuff to arrive so you will have to borrow or walk or ride the bus until then. Also, you can pretty much guarantee a bike you use in the States won't hold up out here. Mine only lasted about 6 months. Unfortunately a good bike here is about $400 but it will be sure to last.
Hopefully other Kwaj folks will pipe up in the comments if there is anything I forgot....