I am putting a new bathroom in basement and want to know the correct wiring. I will be running a 20 amp circuit with a GFI outlet, vanity lighting, and a small exhaust fan. I want to run the fan and lights on their own switches. Can the GFI and others run off same circuit? If so how do I wire it all?

One 20 amp circuit can feed only one bathroom outlet, fan and light, providing the circuit does not extend from that bathroom.

Feed the GFI at the line side ,as will be the two switches, that each will feed the fan and the light.

I recommend you take out a permit for the wiring, along with the plumbing, and possibly a building permit will be needed if structural changes or new partitions are installed.