Ceiling fan lights sparked and smoked causing the circuit breaker to trip. I replaced the light switch on the wall with a 15A 120 vac. This was a very short term fix, about 5min. What is the next step in trouble shooting?

Indicates an electrical short in the fan or fan/light. Fire hazard. Find and repair short or replace fan/light.
Changing switch does nothing for you. Problem is in fan/light.

Thank you!

Would the circuit breaker trip even if the light switch is in the off position? Or could that be a sign of a different problem?

If your power is from the ceiling box the wiring is still hot regardless of switch position. I would almost bet your wall switch has a black and a white wire. That is an indication of a switch loop.

If the fan of the fan/light combo operated with the switch off, that would indicate that power is always available at the ceiling box. If the switch box has only a 2-wire cable, you have a switch loop and power is fed to the ceiling box, as Brian said. If there is a 2-wire cable and a 3-wire cable in the switch box, there is a switch hot leg to the ceiling for the light and an unswitched hot for the fan. Regardless there is always power to the ceiling box.

Kind of depends on what you mean by that statement. If it means that the lights flashed and burned out, it would mean that you had a voltage spike. If it means a single spark occurred, it was probably gone before you could see where it occurred. The tripped breaker indicates that there was voltage spike or a short between a hot and the neutral or the ground. The fact that smoke occurred indicates that the short occurred in the wiring of the fan/light itself. The insulation of the smaller, internal appliance wiring over heating before the breaker could trip.

If this occurred when you turned on the light switch, that indicates that the short is in the light circuitry, the smoke indicating internal to the appliance. If there was a short and you were able to reset the breaker and things worked, that would indicate that something changed or moved relieving the short, at least temporarily. The smoke also indicates wiring was possibly damaged.

OK to attempt to repair, but be certain that you have correctly located the problem and no internal wiring was damaged beyond safe use. Given the price of ceiling fan/light I would simply replace. I have had a similar occurrence and that was my decision.