How do I wire a new light fixture to an old switch that has a ceiling fan on it already? I tried to wire the two whites together and the two blacks to the switch but both lights were dim.

Remove 1 bulb, if other bulb goes out, they are wired in series. Your connections sound?
Correct.

If the switch only has one pair of wires, you have to start at the existing fan. Otherwise, just as Stratmando said, you end up with the lights in series, one feeding through the other.

As Labman says, you might not be able to connect to the switch, you may have to connect to the fan. How many wires are in the box with the switch, what colors, and connected to what?