Replacing subfloor in bath
You got the right idea, but I'd opt for adding a double 2X? between the existing joists at either end of the bathroom span and fill in with an extra joist making your spacing 12 inches on center instead of 24. Add a couple of perpendicular stiffeners, and you will have a good stable base for your new 3/4" sub-flooring.  Tile needs a good solid base or your grout will crack every time.

The vapor barrier is not so important on the floor, however, if you are doing new walls, a properly installed vapor barrier is very important.  There are several methods of adding a vapor barrier, depending on what types of tile backer you plan to use.  Think of it as a system and get all the right layers on in the right order.