EraTOS. It can fit in with the new continuity without being jarringly different from the old. In either case, it isn't constrained by the more fleshed out events of the 24th century.
Setting: war with the Klingon Empire with the Federation operating out of a colonized area.
Ship/Crew: In a departure from other series, they should focus more on the colonists in the sector and the main Starfleet presence would come in the form of the local squadron assigned to explore, aid, and defend the sector.
Dramatis Personae:
-Local Starfleet commander. His concern is balancing his role as the defender of the sector with that of the mediator between local leaders and the Federation. He should be non-human (maybe a Tellarite or other established major Federation species) to make it a little more interesting and to break away from the human commander archetype, and should definitely not be the typical Evil Admiral.
-Federation science officer. He should be the show's protagonist for most episodes.
-Black market trader. Make it an Orion pirate or something. This character is there to provide an alternative to the Federation for the colonists' supplies, travel, and sometimes even their defense. Can have ties to mafia-like organizations that adds conflict into the show.
-Local sentient species representative. They'd actually be better with a few of these. They should be recurring and different from what we've seen before. i.e. they aren't reskinned Bajorans.
How to avoid repeating TNG/DS9
Unlike DS9, the war is simply a fact of life and not necessarily the focal point of the show. If Star Trek was originally a wagon train to the stars, then this is about the settlements out there amongst them. The war can feature prominently from time to time, but the concerns of the characters are as various as their own personalities. The scientists should still be making discoveries and seeking to learn more about the planets they inhabit, and this can either segue into the bigger picture or conflict with it, as necessary.
No one-off planets or alien species that have no relevance outside of their own episode. This only worked in TNG because of Enterprise's diplomatic role. The less new humanoids, the better.
No modern day analogies or allegories.

I'm not interested in seeing another show about terrorists or cold wars or anything like that. Life is more uncomplicated on the frontier, and it frees up space for new plot lines and drama.
To boldly go...or entirely new stuff
I think the lasting appeal of TOS is that it was really not as formulaic as TNG would become. A few of those tropes carried over into TNG, but they should limit forays into time travel, demigods, and space politics. I think Trek showed how worn out its usual method of first contacts and politics had become with the Enterprise episode where the aliens thought that eating was obscene.
Budgets, instead of going to space battles or cosmic wonders should go instead to set design and the creatures. Lets see more of the actual planets they explore...ones that actually have some lore behind them that we want to learn about. I'd like to see the characters getting more use out of their space suits and not just beaming down to the surface for a quick foray. I think that this is one area that Trek lags behind other science fiction simply because they are following a pattern that is a holdover from TOS with 60's studio technology.
Battles that fall into the narrative rather than drive it. They can have an over-arching plan for how the war is going to play out, and they can have engagements here or there to help pan out the story. I'd like it better if, instead of getting it all firsthand from the supreme command, the flagship, or key strategic positions, it should be relayed down to the characters. You can do a lot with characters having different reactions to broadcasts, enemy threats, and new developments in their own sector without having to make it all climactic. Maybe even have the colonists and local Starfleet crews band together to refit miners and obsolete ships to fight against an enemy battle group or something.
I don't see a problem with outlaws, bounty hunters, and the like. The more diverse the various types of characters, the better. As long as they aren't all archetypes or foreheads of the week.
Have at least one writer who knows how to make female characters interesting.