I would have set the series earlier in the 22nd Century, say 2115-2130. All the major nations, that rebuilt after first contact and the war, now have separate space forces; navies so to speak. In the first attempts to make one "unified space fleet (or force)", it was difficult to get experienced, veteran officers and crews. Come up with some nice plan to utilize younger officers, and enlisted personnel, that didn't want to stick to the traditional naval format and had ambitions of their own.
The main ship (I'd actually make this prequel based on a smallish sized taskforce of say 8-10 ships of various designs and weaponry) would serve as an operations hub for the smaller vessels. Exploration of new regions of space would be their key objective, but I would have them also serving as hauling cargo, goods, and on occasion military aid to whomever needed it. They would evolve as sort of futuristic "privateers" for the fledgling unified Earth Space program. I like a lot of the previous mentioned ideas of colonial disputes, pirates (both human and alien), and seeing how their friends and former colleagues view them.
As they manage to meet new races of aliens, they don't hesitate to use force if necessary, and sometimes provoke clashes out of ignorance. They are always willing to trade, with whomever, if it aids their mission. It might be interesting to see this sort of bartering and trading as a means to establish new allies, new foes, and acquiring more sophisticated (or improved) propulsion designs, weapons, and so on. Little by little, they begin to see that there is too much chaos in dealing with strange new aliens in this manner and eventually report back to their superiors at the end of their first mission. They report that they need guidelines and protocols for dealing with other space fairing races that are more neutral.
I think everyone gets the idea and the show just evolves from this point.