Wiring a new GFCI outlet and switch in same box
This is very tricky, because you have both a GFCI device and a switch in close proximity.  It's easy to miswire it so you get an instant GFCI trip when you turn on the light.  

But even before that, there's a fair chance the receptacle is already GFCI protected, and you should check for that. 

If you have one of the common 3-light testers (you want the simple ones with red-yellow-yellow lights), most of them have a GFCI test button.  Try it. If the power goes out, then it tripped a GFCI device somewhere else.  Search for it. 

You can also turn off the breaker that supplies this outlet (since after all, you'll be doing that anyway to service it, right?) First, see if that breaker is itself a GFCI breaker.  If it does, your search is done.   Then, search the house for which other outlets lost power also.  If any of those are GFCI outlets, it's possible the bathroom has been placed "downstream of" that GFCI device, and is receiving GFCI protection from it.  You would need to test further at that point; turn the circuit back on and use "Test" to trip that GFCI.  If the bathroom socket loses power when you do, then your work is almost done. 

If you establish that the bathroom or other plain outlets are downstream of another GFCI device, that outlet needs a sticker that says "GFCI Protected".  The stickers come in boxes with GFCI devices and are also sold separately.