Q: Starter motor problems.

There seems to be lots of juice in the battery - lights, interior lights, radio etc. I turn the key and the lights go dim - there's a single click and some sort of buzzing/whirring noise like a solenoid is discharging. I tightened the negative terminal but that did nothing. The odd thing is that I went back to it a few minutes later and it started fine. I turned it off and tried it again and the same issue occurred. I'm thinking either battery or starter. Thoughts?
My car has an automatic transmission.