There are three black wires, three white wires, three ground wires, and one red wire, all of which are from the supply cirucit. Currently, I capped the black wires, connected the red wire to the black from the light fixture, the white wires to the white from the light fixture, and the ground to ground. However, only one light switch turned the light on. What am I doing wrong?

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There are three black wires, three white wires, three ground wires, and one red wire, all of which are from the supply cirucit.

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That is highly unlikely, only one will be from the power supply.

What you most likely have is a 2-wire cable (black and white) bringing power to the switch box. A 3-wire cable (black, red and white) taking power to the other 3-way switch (and returning) and a 2-wire cable taking power to the light.

You can confirm this by looking at the other switch box. It should have just a 3-wire cable in it.

Assuming that I am correct, in the first switch box, connect the white of the 2-wire cable bringing power to the switch box to the white of the 2-wire cable going to the light. Connect the black of the 2-wire cable bring power to the switch box to the common of the switch. The common will be the dark colored screw (black or brass). Look at the second switch. See which wire is connected to the common there. Connect that wire to the black of the 2-wire cable going to the light. Connect the other two wires of the 3-wire cable to the traveler screws (light colored, silver). It does not matter which wires is connected to which traveler screw.

If you find more than the 3-wire cable in the second switch box, the second 2-wire cable in the first switch box goes to something other than the light.
Connect the two black wires of the 2-wire cables together and to the common of the switch. Connect all white wires together. Connect the red and black of the 3-wire cable to the traveler screws of the switch. Confirm that the same red and black are connected to the traveler screws of the second switch. The common of the second switch will be connected to the black of a 2-wire cable and the whites will be connected together.

If you have something different or need a diagram post back.