I am wiring a BR ceiling fan w/light to a single wall switch.. The new fixture has black, white & GREEN wires while the power source has black, white & RED.. Where does the red wire go? To the black or to the green wiring in new fixture?

The fan black is hot, white neutral and green is ground, Your source is black hot, white neutral and red is unknown and should not be assumed to be ground. Is there a second switch that controls this light, possible even outside the room. Test between the red and white, flipping all switches, is it ever hot?

If the black white and red wires are the only wires in the ceiling box the white will be the neutral. Either the black or the red will be the constant unswitched hot for the fan, and the other will be the switched hot for the light.

The fan light should have a white, black and probably blue wire. Connect the white to white, the black to black of the fan and the red to the blue. The switch should control the light and the pull chain should control the fan. If the switch controls the fan reverse the black/black and red/blue connections to black/blue and red/black.

Edit:

You have 3 threads going. In one of the other post you said you had 2 blacks and two whites. Which is it. Respond here in the "Answer this Question" block. Do not start a new thread.

Was that wall switch always a single switch or were there two switches in one box or another switch somewhere else?