USSF main priority should be to conduct operations that lead to enhancing and protecting the lives of our citizens. This is a broad statement but a fundamental function of our armed forces is to protect our citizens and national interests from threats foreign and domestic under the framework of the Constitution.
There could be different focused offices like that seen in other branches - but designed and enhanced with the exploration, preservation, study, advancement, and utilization of resulting technologies and resources for the benefit of the USA and other countries. We are talking serious R&D into things like clearing the space junk, researching cutting edge propulsion/physics, researching exobiology (including tech to safely study unknown life forms), alternate energy production/storeage/use, metallurgy production in space, advanced life support systems, advanced physics regarding extra dimensional space/generation/uses to enhance safe space travel/exploration etc.
This would in turn make space a safer place to be for USA and human kind in general. Protection from extinction event objects would be good - what if a mission example of the space force could be to identify these objects and conduct operations to mitigate the threats?
The need for multinational support is evident and I think we could do this and benefit all participating nations even by exploiting natural resources in space such as mining asteroids (drone mining vehicle platforms?). The possibilities are many. The potential loss if not tried is scary - remeber, it is not if earth will suffer hits from asteroids but when and how big? If we can build a foundation of safe space operations to explore and protect for the benefit of mankind - what could be better?
Also - in my opinion we should base rank on Navy structure and fighting forces on Naval type organization. I am pro USAF, but I think it would be more natural to think of space as a type of ocean rather than our atmosphere.
Just some opinions. But preservation of our way of life and the worlds' seems like a good priority and start.