01' starting problems!!

Hey guys and gals. Here's the problem: About a week ago, my cavalier (2001 2.2 A/T, 62k mi) started rougher than normal- as if the power was drained. Then the next time it was started, it was worse. I then started it 3 or 4 times and on the fourth time, it wouldn't start at all- Just rapid clicking from the starter. I took the battery in to autozone for the free test and charge. They charged it up, and said it was still a good battery. So I hooked it back up and it fired right up. Two days ago, it did the same thing in my garage. (I started it up about 4 times and it finally wouldn't start...just rapid clicks.) So I had the alternator tested. IT's fine. I cleaned the battery terminals, connections are good. Still- just clicks.... Then yesterday- the car started on the first try with little hesitation. WTF?!?!
I have no idea what it is. It doesn't sound like a fuel issue. The battery and alternator are good. Might it be a starter/solenoid? I don't want to replace anything until I know for sure... I appreciate any advice!

sounds like the starter is the problem to me.

sounds like the starter is the problem to me.

I'm thinking it's the starter solenoid. Would that be a separate issue or do I have to replace the entire starter as well??