Bad starter solenoid, starter, or something else?

I have a 97 toyota camry. Had trouble starting the car 2 months ago, got the battery/alternator/starter tested at advance auto parts, the test said the starter and alternator were fine, but the battery was bad(4yrs old). Replaced the battery that day and the car started fine ever since. For a week or two now, I have had to turn the key 5-10(or more) times for it to start. Sometimes it will start on the first or second try, but usually I have to try starting it for a couple of minutes. When I turn and hold the key to start it, I hear a buzzing sound and it doesn't start. When it eventually DOES start, it either A) starts up immediately when I turn the key all way(as if nothing was wrong) or B) it tries to turn the engine over for 1-4 key turns, and then on the last turn it starts. Once it's started, it runs fine. I had it tested at advance auto parts again yesterday, and this time it said the battery and alternator were fine, but the starter test said "starter amp draw is low with ok starter voltage. Check starter circuit for loose, worn or corroded wiring." (The car also started on the first try when they were testing the starter.) What do you think is most likely the problem (starter solenoid, starter, wiring, etc)? I'm taking it to a mechanic tomorrow, but I was hoping to get a better idea of what's wrong so I can get a semi-accurate repair estimate. Any help is appreciated! :)