The prequel Enterprise is an NX-01, and is a "United Earth" ship. It was supposed to predate the United Federation of Planets. I haven't watched the whole Enterprise series yet, but I think it goes into the formation of the Federation.

I think you might like the series if you start from the beginning. It's kind of fun to watch them, they act like a bunch of cowboys. It is how I would expect us to act, and sort of makes a good segue into the behavior of the cast in the original series. By the time you get to the Next Generation, they're all sort of utopian and idealistic. I mean, the early crews were idealistic too, but in a different way. More sure of themselves, more confident that they can do what's right on their own intuition. The later crews seem to want to philosophize on the "Prime Directive", while the early crews are more cavalier about it. The first crew in the Enterprise Series doesn't even have a prime directive yet, so they tend to jump right in.