Maybe you're going at this the wrong way?
Do you have particular "arch enemies" in mind for your crew? Say, a Klingon commander - smart, sneaky, opportunistic, but still ruled by a code of honour? A Romulan who likes to play "war of nerves" games? The Romulan officer in 'The Enterprise Incident' was - I believe - more than up to match wits with Kirk.
Is your focus exploration, defense, diplomacy -- or a melting pot of all three? This will have an impact on your stories.
Just because Kirk & crew seemed to lurch from one end of known space to the other doesn't mean
your crew will!