A family is the basic unit of our society. Let's learn the expressions for family relationship and for introducing a family.
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Are the sentences grammatical?
What about the use of 'a' ? Am I using the article correctly? Or do I have to omit it?
The sentences are fine except that I'd make "relationship" plural. You're referring to more than one relationship, right?
You might also rephrase the end of the second sentence a bit, depending on what you mean.
To introduce several individuals:
"Let's learn the expressions for family relationships and for introducing members of a family."
If you mean to introduce one family to another family, then your second sentence is fine; otherwise, rephrase as I did above.