[Casiopea]

"in" is short for inside:

inside the covers of the bed
inside the four walls of the bedroom
inside the cabin of the car
inside the four walls of the classroom
inside the four walls of the library
inside the four walls of the school

"at" refers to a general location:

at class (could be inside or outside)
at home (could be inside or outside)
at the library (could be inside or outside)
at the office (could be in your own office or somewhere else in the building)
at school (could be in a classroom or somewhere else in/out of the school)
at work (synonym for 'office', but never "in work")

"on" is short for on the top of the _____ surface:

on top of the bed's surface, on the top of the room's surface (i.e., the ceiling), on the top of the floor's surface, on top of the horse's back,
on top of the plane's / train's inner surface (i.e., its floorboard).

Note, if you walk onto a vehicle, use "on", and if you have to bend down to get into a vehicle, use "in" (e.g., get on the plane, get in the plane, get in the taxi, ?get on the taxi. It means, get on the roof of the taxi. Get on the bus and Get in the bus (from climb inside).

The following already have prepositioned housed within them, so they don't need a preposition:

downstairs
downtown
inside
outside
upstairs
uptown