Space. The final frontier. These are the voyages of the starship
Ok, now I understand. Star Trek is not about exploring the galaxy, or meeting new Alien, boldly gone where no man has gone before. But Star Trek is about a big ship (well 15th - 18th centuries ships in space) with captain and her bridge crews, cruising around the stars and meet Klingon, Romulan, Vulcan and another existing factions. So, why are you refusing "Galactic War / Galactic Political Turmoil" ideas and insist that Star Trek is about "Trek" ?
Enterprise. It's five-year/continuing mission, to seek out new life and new civilisations. To boldly go where no man/one has gone before
If that doesn't mean a mission of exploration, I don't know what does. Trek is about making contact with other races and studying new anomalies, whilst also understanding what it is to be human/sentient. Its as much about the individual discovering themselves as it is the ship discovering new things.
The different series all had a different spin on it, but they were all exploring somewhere new:
TOS - set the standard
TNG - continued it in a grander scale
DS9 - explored the GQ as well as the politics of a single region that they helped shape
VOY - journeyed home across uncharted territory
ENT - mankinds first mission into deep space
The battles and danger they faced were all there to tell stories and put people that we know and care about into extreme situations, so we can see how they conquer them.
Setting it all the way out in another galaxy seems unnecesary, as well as short-sighted IMHO. Unless it was set a century or two in the future, when things are nice and stable in our galaxy adn the UFP has expanded out and routinely travels to the Gamma and Delta Quadrants. Once that was done, then they could spare two or three fleets to better explore Andromeda, as well as ensure the safety and security of their assets in such a remote and unknown galaxy.
There are still plenty of uncharted regions in the Milkyway that could be used for the next series, or better yet, show the hardships of recovering from such a costly war--the true hardwork and sacrifice needed to make the peace work.