Here's what I think will solve my problem:
I want to be able to sing and hit notes. I'd like a program (for almost any platform; in order of preference, Mac OS X, Android, iOS, Windows, Linux) that plays a note as a sustained tone and listens to me as I try to sing the note and hit the right pitch, graphically showing if I'm too high or too low.
I'd like to learn to play an instrument by ear. In finding out how to do it, it is important to be able to replay the song in your mind and tell if you are hitting the right notes, and it helps if you can hum, whistle, or sing it. I can't sing on key, and I can't whistle. Perhaps this is why the whole idea of 'auralizing' (imaging a sound in ones mind) seems strange to me. In any event, starting where I'm at, I think the best step to learning to play by ear is learning to hear if/when I hit a note.
It'd be nice if the application also had a mode wherein it would playback a song, at different rates, and let me try to hit the notes and show me how I'm doing.