First, the 14/2 wire will need to be upgraded to 12/2 wire or you need to downgrade the breaker to 15 amps to protect that section 14/2 wire from possibly overheating/causing a fire.
The vanity typically does not need to be on the GFCI but if the light/exhaust fan is close enough to the shower/tub that you can touch it (if you are not tall enough, image if you were tall enough) while standing in the shower/tub then the light/exhaust fan should be protected by the GFCI. The closet does not been to be protected by the GFCI, but it wouldn't hurt if it was.
GFCI receptacles have two different sets of terminals. One set will be marked line, this is where power from the panel is connected, and other other set is load. The load terminals will protect any outlets downstream that are connected to this.
For anything protected by the GFCI, the neutrals will need to connect to the load neutral terminal. Same as for the hots as well, they need to be connected to the load hot terminal. But in the case of the fan (if it needs to be protected), wire the hot terminal to one of the switch terminals and the other switch terminal to the hot on the fan. The neutral will just be wire directly to the load neutral. If any downstream device does not feed the neutral back through the load terminal on the GFCI, the GFCI will trip.