I am doing well - This is an expression that means that your health is good and perhaps other factors in your life are positive. For example your business, etc. This is how you are doing. Well is an adverb.
I am doing good - This is what you are doing. Here good is a noun. I like to do good to the people around me... for example I say nice things to cheer them up, or I give money to the needy.
Sometimes people make a mistake and say "I am doing good" but they mean to say "I am doing well". It is a common mistake.
Well is an adverb - It usually describes verbs - how you do something
Good is an adjective - It describes nouns - a person or a thing
This pizza is good. (Not 'well')
He made this pizza well. ('Well' describes how he made it, it adds information regarding the verb 'made')
He is a good runner. (Good describes the runner, a noun)
He runs well. (Well describes the way he runs, a verb)
Sometimes 'good' is a noun. Good versus evil.
I hope this helps. :)
Yes, it's becoming more and more common to hear also things like "He runs good", or "He works good", but they are incorrect. This kind of language is slang. It sounds like a brick in the ears of many people... lol
Or things like 'Wow, think of what he must have went through!' Instead of 'Wow, think of what he must have gone through! Yet the same people would never say "I have went"... they know it is "I have gone". And they would never say "He must have was"... Or "He must have took" lol. They know it is "He must have been" and "He must have taken". But somehow you often hear native speakers saying 'must have went'. It's really bizarre.
Somehow some corrupted English becomes widespread, and eventually it starts sounding normal. I suppose that is how the language changes over the centuries.
It would depend on whether you subscribe to a prescriptivist or descriptivist view of grammar and usage. All of the answers you've received here would likely be classified as prescriptivist, as in they view use of language as either correct or incorrect according to a fairly rigid set of rules.
Descriptivists catalog how language is used by various speakers of that language, and have a much wider view of what is 'correct'- things like consistency, systematic use and degree of precision are more important than the sometimes arbitrary distinction of 'right or wrong'.
While I agree with the latter position, it certainly makes things tricky for students, particularly in the early stages of learning. Perhaps the best advice I can think of is 'learn the rules of grammar and usage well before you consider breaking them'.
However, in this particular case, something as common and widespread as 'I'm doing good' is a perfectly acceptable alternative.
@Lydia "I have went", "He must have took" these are very common in American English. As is the construction "If I would have...' instead of 'If I had...' A massive brick in my ear, personally, but these would seem to be very good examples of what I've mentioned above.