Yes, Gulmira, that’s correct English. Despite my previous comment, I like the way you’ve put the shorter phrase (“with you”) before the longer phrase (“some words related to our family”).

Yes, I mean “today I’d like to share some new words related to our family (with someone)"

For example, our teacher says: 

In today's lesson I want to share with you some new words related to our family. 

Is it correct ?

Hi, Gulmira. I’m afraid we absolutely need the “with” (prepositions other than "to" cannot be dropped in English): “I want to share some information with you.” Also, notice that the direct object normally comes before the prepositional phrase in English.