You might think of it this way:
* The verb indicates what happens as a results.
put - place/leave something away from your hand (afterwards it's farther from your hand)
take - get/grab something toward/into your hand (afterwards it is in your hand)
* The adverb shows the way the action is taken.
on - get something "on top of", "close to" something
off - get something "away from", "removed from" something
So, putting them together:
put off -> you place something further away from your hand -> "postpone".
take off -> you get something away from your body into your hand - "undress".
put on -> you place something away from your hand onto your body - "wear".
take on -> you get something upon yourself (you have it in your hand as a result) - "tackle".
The best way to get the hang of it, though, is to study a lot of usage examples.