Should I put vapor barrier only on exterior wall of a bathroom?
The TL;DR -- vapor barriers belong on the outside only

You should only place a vapor barrier on the exterior side of the exterior wall, allowing the assembly to dry back to the inside if water does get in past the tub surround.  I would recommend paperless drywall over cement board for the tub surround, as well -- gypsum board of all types is vapor open (around 50 perms) while serving as an air barrier, allowing drying to the inside without letting a ton of humid air into the wall cavity, while cement board is a fairly severe vapor retarder at 4-8 perms, which puts more pressure on the ability of your tub surround to keep water out of the wall to begin with.

What happens when you try to stick one on the inside?

When you stick a vapor barrier on the inside of a building, all will seem well at first, until you go and turn the air conditioner on.  Then, the moisture in the warm, humid outside air that fills the wall cavity will start to condense on the cold exterior surface of the vapor barrier (assuming an insulated assembly), and you get mold as a result.

A similar problem will happen if you try to insert two vapor barriers in the same assembly -- the inside one will still generate condensation despite the efforts of the outside one to keep all that humidity at bay, and now there's utterly nowhere for the moisture to go!

Of course, there's no point in putting a vapor barrier on an interior wall -- the airspace of a partition should never communicate with that of an outside wall!

More info on why vapor barriers are like badgers can be found in BSI-073, aka "Macbeth Does Vapor Barriers."

Popsicles are for sucking, bathrooms aren't

One other mistake that can be made in a modern, tight house is to slap a "fart fan" in the bathroom and call bathroom ventilation done.  Doing that just means that your makeup air will come from all sorts of random leaks that have no business providing indoor air to people -- especially if one of them happens to be the range hood, or the exhaust of some appliance.

The easiest way to solve this problem is to provide transfer air to the bathroom from a zone that's being blown on, say the basement, if your basement is a suitable transfer air supply, as not all are!  (You'll need to provide, oh, 10% extra makeup air as this is a transfer air setup, as well.)

Of course, if the OP's house leaks more air than a SR-71 on the ground leaks JP-7, an old-fashioned bathroom exhaust fan like the one the OP proposes will serve him fine.

More on ventilation in general can be found in BSI-070, "First Deal with the Manure and Then Don't Suck", by the way.