The Gold Rush era technological advancements made interplanetary travel and colonisation almost trivial - at 1g acceleration it only takes a few days to travel from Earth to Mars. Fusion reactors and Black Box Drives could also be used to terraform and set up very large colonies.
However, the solar system is not exactly a friendly place. Humanity only managed to colonise Earth, Mars and Europa. These three places were strategically placed for mining, hauling, and eventually the war itself. No one really cared about other locations, even though it would turn out other planets and moons in the system were perfectly terraformable.
Contrary to projected outcomes, Earth never became the dystopian hellhole people predicted in the 21st century. Limitless energy (or close enough) reformed lots of third world countries, accelerating them into first world countries in the span of only 20-40 years. It's still the home of the vast majority of humans, and its population keeps increasing.
The red planet was always humanity's dream. Once a big enough colony was set up, most people thought intellectuals and the elite would all move there to establish what they considered to be a newer and better cradle for humanity. This never happened though: Mars instead became the de facto home for miners and haulers working the belt.
Mars is ruled by a form of direct democracy and an office-duty system where citizens are randomly drawn into legislative positions for short periods, similar to how jury duty works on Earth.
The discovery of extraterrestrial bacteria in the subsurface oceans of Europa romanticised Jupiter's moon as a place of great mystery. The colonies of Europa were used mostly as tourist destinations or hydrogen refuelling stations for miners and haulers.
The life-altering experience of standing in the main observatory and viewing Jupiter in its full glory resulted in the formation of a cult referred to as the Red Dots. Some of the cult members were permanently stationed at Europa, and although friendly they still outnumbered the visiting population, so they easily established an unofficial political rule of the awe-inspiring moon.