These are just the most common phrasal verbs with "put." This is not a comprehensive list.

Phrasal Verbs with "put"

to set or place an object somewhere 
or 
to criticize or insult someone (this could also be a noun --> a put-down)
or
to euthanize an animal who is old/sick/injured

to tolerate a behavior or situation
example: We have to put up with the noise until the neighbors move out.

to delay or postpone
example: Can we put off the meeting until next week? 
or
to disgust or bother someone (adjective form is off-putting i.e. offensive)
example: We were put off when he started picking his nose.

to place a person or animal under anesthesia for a medical procedure or surgery
example: The dentist put me under for a tooth extraction.

to dress oneself in articles of clothing or to apply skincare or beauty products
example: He puts on sunscreen before going outside.

to pass a call to the intended recipient
example: Can you put Mary on the phone?

to gain weight or get fatter
example: I put on weight after I broke my leg.

to trick someone into believing something, especially as a joke
example: Don't take it seriously, he's just putting you on.

to blame someone, although the word "blame" is often omitted 
example: Don’t put this on me, the accident was your fault!

to perform
example: The band put on a concert to support their favorite charity.

example: His blood pressure was high, so the doctor put him on a low-sodium diet.

to put (someone) on (a specific medical treatment)

to behave in a certain way to deceive/trick someone
example: The footballer put on an act to run time off the clock. He was not really injured.