My idea, have a rival power compete with the Federation, starships from this other power are trying to get newly discovered civilizations to join them over the Federation and is trying to get claim new planets to take their resources, all so they can be in a good position to attack the Federation at a future date.
You're basically describing the Klingons' role in TOS.
I thought the Klingons were great back then - sneaky, conniving, aggressive yet cowardly, on the UFP's doorstep and doing their best to needle and provoke them. I'd love to see them, or another race, fulfilling a similar role in the new show.
There you go, returning and expanding on this idea from TOS may work as the basis of an ongoing story for a Star Trek show.
I also think you would need a reoccurring enemy commander for this new crew to deal with, build an enemy like DS9 did with Gul Dukat.