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.
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?
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