It's been a while since I ran a Trek game and it'll probably be a while, given I seem to be in a long-term Eberron game (which is a good thing - three months ago I thought my gaming days were coming to an end).
However, I've been thinking a bit about how I would (or hopefully will) run a Trek game should I find myself playing one. Over the past few months I've been reading up on some historical eras I'm not too familiar with (classical periods including the legend of Odysseus), the Vikings, etc. In addition I'm a huge fan of the Napoleonic era (though I'm only now beginning to look at what happened on the continent, having been more interested in the seas..). Also the release of TOS Season One has gotten my mind going.
One thing I notice from the first season of Star Trek is it was in many ways an anthology show with a recurring cast - no alien culture was met more than once throughout the first season - and we never once had a clear visual of any other ship save the Romulan Bird of Prey. The big political game of TNG/DS9 is a ways out in this period. Instead we have a feeling of island-hopping like that we see in Homer's The Odyssey.
The area Enterprise patrols in reminds me of the Agean see of Greek mythology. There are city-states on various small islands, but one never knows when the next island is going to have some witch turn your crew into pigs until the captain can seduce her (man that story is just made for Kirk). But TOS does not have the limitations of Voyager - Enterprise is still in the Federation, just on the fringes of it - one planet might be a place for shore leave, the next one a parsec out might be unexplored.
Another thing I notice is the autonomy with which Kirk operates. Only on rare occasions does Enterprise visit a friendly port - indeed from episodes like "Charlie X" and "The Man Trap" one gets the feeling that Enterprise is in many ways a friendly port herself.
So how would I apply all of this? I think I would go for an era where I would not be tempted to use all of the empires and civilizations established in later eras - they make the galaxy feel too civilized. By the same token, I don't want encounters with friendly folks to be too rare - that rules out Enterprise-era. (And an occasional Klingon is ok too)
Now I'd want to stay clear of established history so I'm thinking of setting this game in the pre-TOS era - something like a decade or so before season 1 of Star Trek. A Constitution class ship on the fringes of the Federation - U.S.S. Odyssey perhaps. What I see is a few sectors of space, each of which has a couple of colonies or outposts but most planets are unexplored. And most colonies are only marginally explored, with some settled prior to the founding of the Federation or the establishment of the Prime Directive. There is a Star Base in the area - not that quick a ride to get there, but it is a port of call if need be. Criss-crossing the space lanes are merchants, speculatie traders, prospectors, and the like - lots of room for the Harry Mudds of the galaxy - and there are certain to be some pirates operating in the vicinity - most are no match for a modern starship, but the occasional surplus Klingon raider might show up from time to time in Orion hands. And the area probably is close enough to Klingon space for me to use the Klingons every once in a while if I so choose. The pirates give some opportunity for some good swashbuckling fun as we fight to drive them away from settlements. And there can be some good espionage too - historically, the American colonies bought a lot of goods from pirates.
As far as visits to planets go, one thing I'd like to do is avoid going through all the trouble of inventing a planet, its culures, life forms, etc. and use it for only one adventure. So planets will be made with a mind for reuse. There are two ways to do this - the first is to have several consecutive adventures on a planet as it is discovered - the first adventure could be the first visit, with later adventures involving learning more about the cultures, setting up an observation outpost to be left behind, etc. The other way is to visit the planet again months later (and the two methods can be combined - a few adventures on a planet and later on having a return visit). I'd foresee the ship visiting each planet for about a month or so.
This is just some rambling from a guy who hasn't played Trek for a while but still things about it. Feel free to use, ignore, comment on, or whatever.