I wrote this a little while ago to answer someone else's questions. Hope it gives you a good overview of what goes on.
Basically, BQC works like this:
-It lasts about four and a half months, with class starting around 08 and ending at 1100, then starting back up again at 1300 and going until 1600. They can cut short your lunch or keep you longer if the discussion needs it, but that'll probably only happen a handfull of times. You take courses in Food Service, Supply Management, Disbursing (read: Navy Accounting), Retail Ops and Leadership Management (read: How to be a Division Officer). These are in turn broken up into modules, typically four or five to a course. At the end of each module there is an exam, each usually lasting about two to three hours. (no lie they really are that long, but everything is open book). There were failures in my class on some exams, but it is very very hard to fail out of Supply School, if not impossible. The instructors are very approachable and the idea is to train and prepare you for the fleet, not weed people out.
You start to think about your orders about halfway through. You will have something called an OLQ board (officer like qualities) where they yell questions at you for 20 minutes and then see how you handle it. That and your exam grades decide your class standing, which helps where you go. When the list does come out, everyone chooses their first choice by class standing. Somewhere on BuPers there is a rough list of when everyone rotates in the Supply Community, so you can probably guestimate what is available, even a year ahead.
As a new chop you will go to sea, or so we have been told by both the detailer and the OP Roadshow (a three hour powerpoint presentation on what your career should/could look like). Although it was talked about on another post, from talking to other instructors and from the Detailer roadshow there are no operational billets in aviation at the JO level that I was ever told about. Coming out of the BQC you will have the choice of a ship, a sub, or maybe SeaBees.
Once at sea, you will probably be assigned either as a disbursing officer or food service officer. They both have their own pluses and minuses. Disbo is the guy with all the accountability; you have to be dead on in your money counts and such or else. (Else being a one way ticket home.) Food Service is always in the spotlight, since everyone is going to pass through at least once. Disbo is more low key, everyone will know the FSO, which can be good or bad. On Subs and Sweeps there are only one Chop, so it will count as a department head tour for you. Its high responsibility but high reward; your DH ticket gets punched early, something that they will look for if you want to advance to O-4.
Any other Chops out there lurking can chime in if I screwed something up or to throw anything else in. Disclaimer: all this is coming from a (very) junior officer with just a foot in the door and should be taken with a grain or two of salt. Hope this helps!