Susan-Marie,

Usually the last note of the song, particularly uncomplicated traditional songs, is the key of the song. This is called "resolving" the music. This is a guideline, not a hard and fast rule. The piano method you describe is good also.

Sharps and flats not part of the key signature are called accidentals. If a note is consistently sharp or flat, that should be considered in the key signature. If it's only occasionally sharp or flat but mostly natural, it's an accidental. Of course a natural can be an accidental also, if the note is sharp or flat in the key signature. Complicated enough?