I have a Hampton Bay fan with remote - I know the basic wiring installation however coming out of the ceiling I have a Green, Black, White, Red wire.

I know from the fan to the ceiling to connect: Green - Green, Black/Blue - Black, and White -White. But what do I do about the Red wire?

You probably have 2 wall switches, 1 for the fan and 1 for the light. With the remote you don't need 2 power leads. Cap off the red and stick it back in the box.