Doing Good and Doing Well are both common in English. They can have slightly different meanings but in general, they both mean that life(especially health and wealth) are in a good place.
In English grammar, Good is an adjective and Well is an adverb. An adjective describes a noun and an adverb modifies a verb. In this case, the correct grammar is “doing well” as the adverb “well” modifies the verb “doing”(it adds more meaning)
Many native English speakers say “I am doing good” but note that many people who enjoy correct grammar will say” I am doing well”.
We use adjectives with verbs that express states of being(be, feel, seem).
For this reason, we use good with these verbs.
I am good.
You look good.
That new band sounds good.
We use well with verbs of action because we need an adverb to go with the verb.
I am doing well
You play the guitar well.
You sing well.
Is well an adjective?
Well is usually an adverb but well can also be an adjective when we are talking about health.
For this reason, we can use well with verbs that express state.
I am not well.
I don’t feel well,
You look well.
In all the examples above, we are describing the state of someone’s health.
What does doing well mean?
Doing well can mean that things in general are in a good state but we can also use it in more contexts. Doing well often has the meaning of achievement or health.
He is doing well. (He has a lot of money).
She is doing well in school.
How is your father?
He had a rough time but he is doing well now. (His health is improving)
Are you feeling good or well?
It is possible to use both good or well in this situation because in this case, we are talking about health, and for that reason, we can use the adjective good or well.
You can’t use feeling well if you are not talking about health. When you say “I am feeling well”, you often imply that you had a previous illness.
What about “do good”?
There is an expression “to do good” It means to act in a moral way especially helping out with a charity. In this expression ”good” is a noun.
I hope I can always be helpful and do good.
Think of this expression as one noun “do good”. There is also an adjective used to describe people which is do-gooder. This is often used in a derogatory way.
She is such a do-gooder.
Is well better than good?
Well is not better than good. The main difference is that they are used differently.
I received a good result in my exams.
I did well in my exams.
Both good and well suggest that you received a high score in the exam but not an excellent score.