It is reasonable to assume there will be a fair amount of technological infrastructure installed in the solar system prior to post-human colonization. What is often overlooked is that automated and cybernetic machines will be the vehicles and industry occupying space and objects in the solar system.
The post-humans may not colonize various bits of solar real estate, but they may exploit their resources via self-aware and self-directed machines. Once they created more than sufficient infrastructure, the post-humans will take up residence in the more desirable parts of the solar system.
Post-human colonization is likely to proceed in terms of accessing the mass and energy they need to further construct and expand their occupation of the solar system.
The Moon and near-earth objects (NEOs), basically asteroids, will be initially the sites of solar power arrays, automated mining machinery, and mass-drivers to ship out material. No post-humans need be on site. A combination of AIs and remote control can do the job.
Next go inwards for energy and more mass. Mercury is believed to be rich in minerals and metals. Close to the Sun there will be solar power to burn. (Apologies for the demotic turn of phrase!) Venus can be harvested for carbon-based material. Floating structures will be used both as infrastructure and possible habitats for post-humans. The view will be spectacular.
Both planets are energetically easier to get to, than going further outwards.
The next phase of colonization will be the planets Jupiter and Saturn and their attendant moon systems. The first wave will always be AI controlled systems to establish the basis of later industrialization, habitats, and colonization.
Both planets and their moons are rich in valuable and usable resources. Water, deuterium, carbon, oxygen, nitrogen, aluminium, and silicon. Also, it's hard to imagine any colonization would ignore Titan as it has to be the most Earthlike world, outside of Earth itself, anywhere in the solar system.
One problem in deciding the order of a post-human colonization of the solar system would proceed is that much of it will happen simultaneously and there will be considerable overlap in their settlement. This could happen to such an extent that the colonization might not proceed in any specific order.
For example, the exploitation and utilization of asteroids will start early with NEOs and then gradually continue to embrace the asteroid belt proper. It may be continuing to expand while the gas giants are being colonized.
Also, there is a degree of undecidability in knowing the impact of the plot driver for the Great Goal of an interstellar mission. But yourself as the author will have to make that decidable.