First, usually (or always, according to some, though I’m not part of them) no comma before et introducing the last element of an enumeration. In this specific case, leave it out. Your first version is correct with two minor corrections:
Son père, sa mère et ses frères vont à la plage / se rendent à la plage.
Your number 2 is not usable in any case, since the possessive is singular and is to be distributed to many things (persons in this case, I guess).
Your third option, though incorrect (edit: it turns out I was convinced about being wrong here, this form is indeed used in formal speech, but very less so in casual speech about people going to the beach), is not as offensive as the second. I do not recommend you use it as is, since it not very French-like, but it could serve as a base to go through your enumeration a bit quicker, with the price of introducing it with an extra bit of information:
Sa famille, père, mère et frères, va à la plage / se rend à la plage.
The first part introduces the family, the subsequent enumeration details it. In both versions, the verb (va, se rend) is singular, since it describes the action of the singular family as a whole, and not of its many individual components.