At one point in early TNG Worf refers to something in “a nearby quadrant”. It seems there are smaller quadrants as well as the four quadrants of the galaxy.
Deep Space Nine, of course, was about a wormhole to the Gamma Quadrant, again far away in the galaxy.
Torpedoes are regularly used at warp speeds, and obviously have some internal propulsion.
Again from TMoST:
Assigned to operate in a far-off sector of the galaxy, the ship’s duties include: scientific investigation and reconnaissance of previously unexplored worlds, providing aid and supplies for Earth colonies, diplomatic courtesy calls on alien planets, and the enforcement of laws regulating commerce between Federation members. Except under orders from a higher authority, the Enterprise *never leaves its assigned area. *[Well, almost never, anyway. :rolleyes: ]
When you consider it, the Enterprise * is doing the same kind of job naval vessels used to do several hundred years ago. In those days, ships of the major powers were assigned to patrol specific areas of the world’s oceans. They represented their governments in those areas and protected the national interests of their respective countries. Out of contact with the admiralty office back home for long periods of time, the captains of these ships had very broad discretionary powers. These included regulating trade, fighting bush wars, putting down slave traders, lending aid to scientific expeditions, conducting exploration on a broad scale, engaging in diplomatic exchanges and affairs and even becoming involved in such minor matters as searching for lost explorers or helping down-and-out travelers return to their homes.*
This wasn’t firmly established until ST II: TWoK. Prior to this, they were defined as “energy pods of matter and anti-matter contained and held temporarily in a magno-photon force field.” No mention was made of a sabot, much less a propulsion system.
While we’re on the topic, how can the phasers work while the ship is traveling faster than the speed of light? Wouldn’t they risk shooting themselves down?* :dubious:
*I’m being facetious here. 
This is all hard science fiction. Science says that FTL implies paradoxes, and Star Trek has FTL, so it’s scientifically correct that Star Trek contains paradoxes.
If there’s nothing wrong with you, there must be something wrong with the universe!
Is Alpha Centauri in the Ninth Quadrant, or is it just the starting point for the mission? (ie “You will patrol the North Sea, leaving from Virginia”)
I always thought that quadrants were divided into sub quadrants.
“Big fish eat little fish,
Until bigger ones can gram them,
and so on, and so on,
it continues, ad idfinitum”
Check out John Scalzi’s Redshirts, if you haven’t already, for a clever, and occasionally even touching, meta take on what it would mean to serve as a junior officer on a ship like Kirk’s Enterprise.
The fan understanding I’m aware of is that the original 3 seasons plus the 2 animated seasons cover the entire 5 year mission.
Also, while they may start out at the edge of the galaxy, they make it back to Earth by the end of the second season for a time travel mission.
Problem is, Star Trek also posits that Warp travel gets around those paradoxes.
So they just bring in more with black holes and such.
I’ve been re-watching TOS in sequence (at least the sequence on Netflix, which isn’t exact but is close), and there doesn’t seem to be any progression in any direction. They’re all over the place, to the extent that can be determined. There’s no real continuity in TOS.
Yeah, but the problem with that is that they tossed that stuff in the oubliette within just a few shows of starting. Which is why I ignored this and talked about what the actual episodes talked about them doing.
You realize I was joking about it being hard sci fi, right? Soft and gooey with some harder lumps in it, maybe.
Which episodes aren’t covered by the description in the first paragraph of terentii’s quote? I agree that many episodes involve activities not covered by the voiceover at the beginning of the show, which only covers exploration, but the examples you gave fit the description above (with the exception of taking Spock to Vulcan, which was done by Kirk in direct defiance of his orders to attend a planetary president’s inauguration ceremony). They explore but also provide services in aid of the Federation’s objectives, including bringing supplies to colonies, response to emergencies, support for diplomatic missions, etc.
Showing the flag, and rattling the sabre, as someone stated above, much as RN frigates did in the 19th century.
On edit:
Oh, delivering breadfruit and quadrotritacayle.
My second favorite line in Galaxy Quest is when Jason tells the others (who have just been teleported aboard) that they are in the 23rd Quadrant of the Gamma Sector. 
I’m fairly sure that was a deliberate slap at the TOS writers continually confusing the two terms.
Any guesses what my favorite line is? 
Guy Fleegman: Did you guys ever WATCH the show?
Stranger
No, but I’d put that in the Top 10 someplace. You did get the right character, though.
Guy Fleegman: HEY! Don’t open that! It’s an alien planet! Is there air? You don’t know!
There are so many good quotes in the film it is hard to guess, actually. But casing Sam Rockwell in Vice looks like brilliant casting. Nobody plays charmingly clueless like he does.
Stranger
When there is some small hint at continuity, it gets mentioned in the Mission Log podcast for the TOS episode. The Organian Peace Treaty is the most famous example, but there are a few others. Not many, though.