Basically, they'll tell their chain of command they want to leave, and the squadron AOC will start the ball rolling. There's a series of appointments and tasks that need to be completed for a cadet to leave. For a voluntary resignation right after BCT (most people call it the Form 34 process), it would probably wouldn't take more than a couple of weeks at most. If there's a medical issue and becoming a turnback is an option, that can extend the process. I've seen Form 34 departures happen on the same day the cadet wishes to leave (very rare), and I've seen them take over a month.
The process takes longer if it happens during the semester. For example, if a cadet decides to Form 34 right before final exams in December, they will almost certainly be required to return after winter break to finish outprocessing. If there are pending disciplinary or other issues, that extends the process too.
The AF won't pay to send them home and won't collect any money, but will keep some of the issued gear. Typically, until a cadet physically departs USAFA, they will be required to attend mandatory activities, classes, and formations (mostly for accountability.) So, yes, it is likely that a cadet who expressed a desire to outprocess at the end of BCT would be present for the Acceptance Day parade.
Some of this info may not be 100% current if the process has changed recently.