93 Corolla starting problems

I just replaced an alternator on my 93 Corolla. It worked fine for a while, and now I'm having problems again. I had the alternator checked, and it's charging fine. My problem is that sometimes the car doesn't want to start. No clicking of a dead battery, full charge. It just doesn't engage at all every once in a while. And when it does engage, it's after a long hesitation before turning over. Could it be the starter? And should I replace the entire starter, or just the solenoid?

Sounds a lot like a dead starter motor. Take it to an auto electrician first - it may just be some wiring/grounding. :)