97 Corolla (sometimes) won't start

Thanks. The cables seem good, although I’ll take a closer look. I don’t have an anti-theft system. Any way to test/check the ignition lock? I’ll look into the starter solenoid as well.
It started fine this morning (19 degrees and snowing, had been outside all night). But yesterday evening it gave me a scare: first two turns of the key: nothing. Third turn I held it all the way for about 6 seconds and then it started right up.