Vapor barrier on heated entertainment shed
The insulation goes in the stud bays between the studs. There are several options for insulation some of which are fiberglass bats, spray foam or solid foam  sheets.
Once the stud bays have been insulated then you cover the entire wall with a sturdy visqueen or plastic. 4mill will suffice, but 6mill can be used.
I like to staple the visqueen around the perimeter and leave enough so that it wraps around to the adjoining wall and ceiling and then staple and caulk that wall or the ceiling’s plastic on top of that.
For added Insurance  you could run a bead of caulk all the way around the perimeter of the wall before you staple the plastic on but you do not need the caulking on each stud.
If you puncture the plastic anywhere make sure you repair it with a suitable plastic tape.
Do a search on YouTube for how to install vapor barrier in new construction and you will Have much information at your disposal.
Caveat: Not all information found on YouTube is correct so be sure to watch several videos,  And research in other places.