If the lights are OK, then I would check a few more things.
Check to see that other accessories work ok, like the radio and the fan.
If you can run the fan, and the lights OK, then likely the battery, and the battery connections are not a problem.
Do the engine cluster lights check OK, which would mean that the oil and check engine light come on with the ignition turning on?
If yes, then your ECU (computer) is likely working OK.
If everything works to this point, I would be suspicious of either the solenoid for the starter motor, or possibly the ignition switch. Usually there is a fusable link or a large fuse for the starter solenoid. With someone listening under the hood, activate the starter, and have them listen for a click. If no click, the ignition to starter solenoid is likely the issue. There could be a failed starter, but they usually do not fail suddenly.
The make/model/engine for your car would help better guide you to a starting solution.