I've read some people experience fever or flulike symptoms after having one of the covid vaccines, particularly the second shot. how soon after the shot are these effects most likely to occur (12 hours? 24?) and how long do they take to go away?

Due to the overwhelming availability of the COVID-19 virus, receiving the vaccine shot (first or second) may occur after coincidental incubation of the virus at the time of vaccine administration, which produces symptoms as described, since relative immunity does not occur simultaneous to the administration of the vaccine, but only weeks later.