Hi Sasha,
There are a few ways to express your idea.
1) You will need to introduce everyone else (i.e. my mom, my son) before yourself (i.e. I).
"This June, my mom, my son and I [visited] Sasha's mother in the village for [two] weeks." (correct)
"This June, I, my mom and my son and I [visited] Sasha's mother in the village for two weeks." (incorrect)
2) You can also use a prepositional phrase using "with" to show accompaniment.
"This June, I visited Sasha's mother in the village with my mom and son for two weeks."
"This June, I visited Sasha's mother in the village for two weeks with my mom and son."
3) Your suggestion of using "I together with my son and..." is more suitable if the sentence is written instead of spoken: If your focus is mainly on yourself, you can use appositives.
"I, together with my son and mother, visited Sasha's mother in the village for two weeks this June."
In this case, "together my son and mother" is an optional piece of information -- an appositive.
It sounds "cumbersome" to read though grammatically sound.