This is difficult, because there are many different situations and these prepositions are used in many different idioms, some of which seem contradictory.
There are some loose rules, however.
You might use "at" in the same sense as 在, indicating a location.
"Where is he?"
"He's at the doctor's [office]"
If I am seated at the table, I am seated close enough to eat.
"In" is used (usually) in the same sense as 里, or inside.
"Where is mom?"
"In the kitchen." [Inside the kitchen]
"Where are your keys?"
"In my purse."
"on" is often used in the same sense as 上, but not always.
"The book is on the table." [on top of the table]
These rules often hold, but not always. We say we are getting "on" the airplane, when in fact we are getting "in" the airplane - at least I hope we are.
This is not a complete description by any means, but it covers most of the area you mention.