It somewhat depends on what precisely you mean. Of your options listed, "job title" seems the most sensible and most likely to be encountered.
"Job position" is not a combination I have heard of, but you can easily guess its meaning.
"Position" is sometimes used, but I would expect it more likely to be used in conversation than on a form
Example:
"What is your position at this company"
"Post" has an older, more British or military sound to it, so probably not a good fit.
If you are asking what the person does now at their current employer, job title is good. If you want to know what they do in general for employment then profession or career might be suitable.
"Stuff" is informal, especially when used as a de facto pronoun. "Stuffs" is normally used as a verb, not as a plural pronoun.
"Thing" is a standard pronoun, so it can be used in both formal and informal writing. "Things" is a normal plural pronoun.
I am going to give my children food, clothes, and other things.
sounds formal. "I am going to", "give", and "children" are all formal choices of words.
I am going to give my children food, clothes, and other stuff.
sounds informal enough to almost seem incorrect. Using "kids" instead of "children" would make the sentence more consistently informal. Using colorful metaphors (like "put clothes on my kids' backs" or "put food in my kids' bellies") would also be more consistently informal than "give my children <things>".
In informal speech, it is normal to use contractions for some words. Even in informal writing, it is normal to transcribe these words using the full words. Thus, your written "I am going to give" might be spoken informally as "I'm gonna give".
Best Answer
When using that phrase as a title, both variations are valid. There is, however, a slight difference in meaning, I think.
For me, My father and I concentrates a bit more on the things you and your father experienced together, while My father and me emphasizes about your relationship.
These are felt meanings, though, so both may be used in any case.