Set in the 23rd century , the first Star Trek follows the adventures of the starship Enterprise and its crew , led by Captain James T. Kirk ( William Shatner ) , his First Officer Mr. Spock ( Leonard Nimoy ) , and his Chief Medical Officer Leonard McCoy ( DeForest Kelley ) . The original Star Trek series follows the adventures of the starship Enterprise and its crew , led by Captain James T. Kirk ( William Shatner ) , first officer Mr. Spock ( Leonard Nimoy ) , and chief medical officer Leonard McCoy ( DeForest Kelley ) .