You are absolutely correct. "How do you do" is an old fashioned introduction and is an obsolete synonym of "hello", and consequently the proper response is "how do you do?".
Mr Darlington: How do you do, Mrs Windermere?
Mrs Windermere: How do you do, Mr Darlington?
This exchange is exactly equivalent to the more modern (and at the time, more vulgar):
Mr Darlington: Hello, Mrs Windermere.
Mrs Windermere: Hello, Mr Darlington.
Note that the phrase "how do you do" has long since gone out of fashion in British English, and was never in fashion in American English.
Generally speaking, native speakers in both British English and American English now use the phrase hello instead, although how do you do does still persist in some formal settings in British English, generally during formal introductions, and lives on as howdy in some dialects of American English (and like how do you do, the correct response to howdy is howdy).
Don't be surprised if using this phrase that you receive odd looks - the phrase is excessively polite, and may easily be mistaken (even by native speakers) for the more common introduction "How are you?", to which the proper response is "I am fine, thank you, and you?".
When saying goodbye, take it easy and take care are virtual equivalents. I wouldn't construe either one to mean anything more than a polite yet standard way of saying goodbye when two people depart from each other.
I also would like to know what situations I can use the phrase "Take it easy."
There are several uses of that phrase besides the standard goodbye. In context, it can be used when:
- you try to calm someone down (Bob was about to get in a fight when his friends told him, "Calm down, take it easy.")
- you want someone to be cautious (Jill was teaching her daughter how to drive, when they approached a sharp turn in the road. "This road takes a sharp turn up here," Jill said. "Take it easy on this curve.")
- you want to tell someone they should embrace a more carefree lifestyle (Ted said to his friend Janet, "You stress out too much; you should just learn to take it easy.")
- you want to help soothe or calm someone's anxieties or emotions (Brenda broke down in tears when she told her husband the bad news: she had just lost her job. Her husband embraced her, and tried to offer some reassurance. "Take it easy, honey," he said. "Maybe this will work out for the best somehow.")
Similarly, "take care" is usually just a shortened form of "take care of yourself," which is why some folks may say that to you when you're not feeling well, either physically or emotionally.
Both phrases often have undertones of empathy, although "take it easy," usually means to slow down or relax, while "take care" means to get well, or remain in good health or spirits. As I said before, though, when either of these are used as a substitute for goodbye, there's a good chance the speaker used one instead of the other only arbitrarily, unless something earlier conversation might have changed that.
Best Answer
Whether or not one hears how do commonly may be a matter of locale. English is pronounced, mispronounced, and generally mauled differently in each corner of the world where it is spoken.
I have heard the greeting how do? at least three times since yesterday noon in rural NE US.
It is, as others have said, a shortening of "How do you do."
Although one can parse the parent phrase into a grammatically complete (if slightly ridiculous) sentence, in which the subject you performs the intensified verb do do which is modified by the adverb how, the exercise in doing so would be purely academic doodoo, as in all probability few if any greeters have truly wished to know how, in fact, someone else does do, did do or has done doing since the 18th century.
Instead, the phrase falls into the category of phatic expressions. All hail Wikipedia:
Like other phatic expressions (e.g. Hi, Hey, Yo, and Heighdy-Ho) it neither needs nor wants grammatical analysis.