Yay. So let's do this! (That's the required modern lingo.)
Everything about my new timeline centers around the Eugenics war.
Before the Eugenics Wars, Earth had multiple nations colonizing the stars. Humanity actively altered itself to fit into these new environments, often after the colonists had arrived. This active altering came to be called Eugenics, even though that's a misnomer. (This is why almost all aliens look like humans and can interbreed.)
The Eugenics Wars happened on Earth, as some folks made themselves superior. Many human groups fled. The number of unapproved colonies went through the roof. All sorts of bad stuff happened. Thanks, Kahn. All colonies severed ties with the mother planet. About the same time, the world of Vulcan ripped itself apart as the two factions fought for dominance. In the end, the Romulan faction retreated into deeper space.
After the Eugenics war, Earth banned the use of Eugenics. With Vulcan and Andromeda, they formed the United Federation of Planets. Rather than a galaxy bound to Earth, they would be bound together in a voluntary union. (This explains why there are so many planets, yet humans, who only have one planet, dominate the Federation. They just have more people than any other race.)
During the war, the Klingons feared that Kahn would win the Eugenics war, then come after them, so they used their tech to breed a soldier race to defend them. Thus, the high born Klingons looked very human, but the soldier race looked very fierce.
Once the UFP was established, it was time to go out there and find more planets to bring into the Federation, but as so many colonial records were destroyed in the wars, or never registered in the first place, finding them gets a bit tricky. Thus, you need to send out missions.
All would look good if not for those pesky Romulans, who had begun their own star empire, or the Klingons, who also thought that one was a good idea. And some in the Federation would come to see the UFP as a sort of soft-empire, but one just as ruthless. Her enemies knew better.
Given the size of the UFP, soft power mattered more than hard. Rather than build pure warships, the UFP put its energy into multi-roled ships, able to explore, transport, and fight as needed. (This is why smaller alien warships are often able to fight the much larger Federation ships to a standstill.) Indeed, the Federation's excess human population provides it a surplus of manpower and colonizing power while other civilizations are still working at building up their core planet. Although not absolutely stronger than any of their enemies, the UFP can overrun their opponents through sheer force of numbers. This is why the Klingons prefer capturing existing human colonies, as they just don't have the population to establish them themselves. The Romulans also use force to bring planets into their empire, but like the Romans before them, understand the limits of force, letting those colonies manage themselves.
That's it in a nutshell. Nobody paid me to do this. Nobody's going to pay me in the future for this. Make of it what you will.