Martin P. Robison performs Slimey. The first time I heard Slimey speak was in the episode where Slimey returned from the moon. He might have spoken before that, but I don't know for sure.
In 1985, Golden Book Home Video released two Sesame Street videos with various Sesame Street storybooks made with limited animation, and one story that was included was The City Worm and The Country Worm. Slimey spoke full sentences in that story (although in the video version he was called Squirmy). He was obviously not voiced by Martin P. Robinson in that video, but I don't know who did his voice.