And yes, a trill starts on the note above (a plain trill that is). Some mordents are notated to start on the note below. Some fancier ones may also have the next to the last note of the trill on the note below (this will be notated).
One other item. Ideally, the trill is in the meter, usually twice as fast as the smallest notated division in the measure (for instance, if the small divisions in the measure are eighth notes, the trill will sixteenths). Occasionally this is varied with the trill being four times as fast, but that's not common.