I have a three room apartment in an old house with 60 AMP service
coming in from the main. Currently the kitchen has 15 AMPS going to it
and the other two small rooms also have 15 AMPS between them.

I need to re-wire my bathroom for a light fixture, single pole switch
and a GFCI outlet. I was thinking to get a new 15 AMP breaker and run
a 14/3 gauge wire from the box to the bathroom.

Can I get away with a 14/2 wire instead of 14/3?

With the power off. My proposed connections would be:

Place the 15A breaker in the box in the slot below the second breaker.

Take the 14/3 or 14/2 band and slot it through the box and then connect
the white wire to the white column in the box.

From the 14/3 or 14/2 band take the ground wire and ground it to the
box.

From the 14/3 or 14/2 band take the black hot wire and and connect it
to the 15 A breaker via the screw.

FIsh the 14/3 or 14/2 band below the subfloor and then to the area of
the switch or outlet in the wall.

The metal boxes are in place for the switch, GFCI outlet and the light
receptacle.

Now how do I connect the remainder?