The ceiling box had a black wire, a white wire, a red wire, & a bare ground wire. I connected the black to black, white to white, ground to the ground on the fixture strap, & ignored the red wire. The light now works but not with the flip switch. It works with the round switch when it is turned to the third position. Should I have used the red wire or do I need replace the wall switch with a standard dimmer switch?

The fan was supplied through 2 wires, 1 for the fan and 1 for the light. Switch the red for the black and see if the fan is now controlled from the wall switch. You can also pull the switchplate off and see what wire is connected to it. If you want to, you can install a dimmer when you find the right wire.

The red wire is connected to the standard switch not the round one sounds to me the round one was for the fan motor. So to answer you yes connect the red wire to the new lights black wire that will fix it. You also could change out the round switch too. Make sure at the light you wirenut the unused wire.