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.
Okay, I really like the idea of the combined space navy of Earth's government. It'd incorporate that into my ideas. It would allow for many different type of ships and it was have the awesome international flare that this very...human series would really need.
The question is why would Earth have such a large naval force? True it is 60 years after WW III and that might be enough time...to rebuild after the Post Atomic Horror. I always figured that the PAH lasted maybe a good generation. Those people seemed well set in that system.
Usually a culture that rebounds that quickly as a planetary group does so because it has to. Are humans paranoid by this time?
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.
One of the things that was left out of my ideas and Enterprise too was that there should be some time to deal with the situation on Earth and the continued rebuilding and showing of good faith between the nations and how fragile that relationship is...we can't give earth the look of being united if it wasn't united a few Decades ago...There should still be contention and strife and even the danger of war..