I am designing a future history setting where humans develop the technologies of genetic engineering long before those needed for terraforming, yet at the same time they have access to some form of FTL travel (handwaved as a product of transhuman genius). They set about building outposts on worlds throughout the Orion Arm for various reasons (industrial, scientific, religious, etc.) but in order to do so the settlers agree to genetically engineer themselves and their descendants into a form conducive to life on world surfaces with higher level of background radiation than Earth's surface. My question is what would these radiation-adapted colonists look like? What adaptations would be necessary for them to survive on a world that has similar characteristics compared to Earth (mass, orbit, etc.) but which receives more radiation from its host star?
Needed adaptations would be on a biochemical level. They probably will not show.
If your future folks go around in the buff, then adaptations to the epidermis would be relevant. And I heartily implore you to write such a fiction. But your future folk would likely be the aforementioned indoorsmen, and so the relevant radiation would be energetic radiation that can penetrate structures, clothing and epidermis - short wavelength X-rays and gamma rays and charged particles aka cosmic rays.
These can cause damage anywhere in the organism. Lots of radiation in a short time affects proteins and creatures so damaged are cooked. Long term exposure to ionizing radiation causes increased rates of cancer.
Epithelial and mucosal surfaces are more cancer prone and so have more resistance mechanisms to cancer. These are the interfaces between body and environment - skin, gut lining, bladder lining etc. They turn over faster (and so die before bad mutations can take place) and are more exposed to toxins. Increased amounts of ionizing radiation do not lead to increased rates of common epithelial cancers like breast or colon cancer (possible caveat for children so exposed).
Deeper structures have less intrinsic cancer resistance mechanisms and these are the structures that develop more cancer where there is an increase in ionizing radiation - sarcomas of muscle and bone, and cancers of the brain.
Radiation resistance is evolutionarily flexible. If you have more than you need it is a waste of energy - like having plate mail 10cm thick. If you have less than you need such that it impacts genetic fitness there will be evolutionary pressure to get more.
You can't have cells in the brain turn over faster (as might be adaptive in the gut or skin) because those immortal cells don't turn over at all. Those cells need to express more DNA repair mechanisms. Your people engineer themselves to get more cellular DNA resistance. That might be as simple as up regulating DNA repair mechanisms in the relevant cells - or maybe in what on the organismal level is a post-scarcity world (these folks all have plenty of food), up regulate in every cell.
Unfortunately for your fiction, that does not much change appearance. Improved repair capacity of brain cells will not render your characters into blond beasts. If that is your dream, your characters will need to work out, and they will need peroxide.
Any substantial increase in radiation will need adaptations that are invisible. However if the radiation increase is minor, there are no other factors, and evolution can run, over thousands of years, there may be a physical change.
We'll get shorter. Even on earth, The taller you are, the higher your risk of cancer.
Every time a high energy particle strikes you, it has a risk of doing damage. The smaller you are, the less impacts you receive. The less impacts you receive, the less cancers you get. The less cancers you get, the more likely you are to raise children to pass on your genes.