Once one Englishman said me ‘to engage’ means – ‘get into contact’ in any its meanings and after that I began using it easier. Unfortunately dictionaries give dozens meanings for a word and learners get stuck to memorize a lot of different meanings for one word. In this system could you say what is the core, 1-3 main meanings for ‘to put’.
The primary meaning of put is "to move into a certain position".
He got home and immediately put his wallet and his keys on the small table by the door, where he wouldn't forget them.
As with "get" there are many phrasal verbs and idioms that use put, but (to some extent) all rely on this basic underlying meaning. For example:
As the clock struck nine, the children obediently put away their toys and took out their schoolbooks
Put away means "place an object in its proper location", or "place into storage".
At the meeting, the young manager put forth a bold plan to reorganize the production line, in a way that he said would increase efficiency by twenty percent.
Put forth means to "offer for consideration". Figuratively, the image is like placing some object in front of the intended recipient.
Arthur wouldn't stop putting down his boss, saying he was completely incompetent and didn't deserve the job.
This meaning of put down is to "belittle", "criticize", or "humiliate". Figuratively it is like placing that person in a lower or reduced position, so that you can "look down" on them.
And many other phrasal verbs and idioms. Of course, there may be a few expressions where the connection to "place" is more tenuous -- such as put up with -- but for the most part this should be a good guide.