How does The Expanse solve the big problems of initially settling on Mars?

Radiation: the domes were built to keep out radiation, and cities were built underground while terraforming was ongoing.

Energy is already wholly satisfied by fusion reactors.

From Cibolas Burn:

Prologue quote confirming no magnetosphere was set up on Mars:

"It's going to be generations beefore anyone walks on the surface here! We don't have a fucking magnetosphere!" - Bobbie Draper

Chapter Two quote from Elvi's thoughts, emphasis mine:

When the first colonies had begun on Mars, the perimeter domes had been a question of survival. Something to hold in air and keep out radiation.

So you can see that the domes were in fact built to keep out radiation, and the ones they brought along for the expedition to Ilus/New Terra were very similar to the ones originally used on Mars.

Chapter Twenty-Five, emphasis mine:

"The newer areas, the ones built after we had some idea of what folks need to feel happy, were built different. No more narrow stone corridors." - Alex

Epilogue, Avasarala's thoughts:

the entire city--the entire planetary network--was built like a fucking tomb. The absence of a magnetosphere had made Mars' first priority protection from the radiation.

So in the initial build on Mars, they tunneled into the ground, and continued doing so to hide from radiation -- the rest of that quote describes how they figured out how to make it feel less horrible. It should be noted that they did the same on Ganymede by burrowing most of the place underground, though with the exception they weren't attempting to terraform.

The rest of this is speculation but:

Food is shown to be pretty simple to make with rice and yeasts and mushrooms and other such stuff being the staple of outer planets diets, it's likely, though I can't prove it, that they used that and imports from Earth to start -- at the time of colonization they were still a UN colony so they would have a vested interest in keeping them alive, and since they were a portion of the population from Earth, the capacity to do it.

Water is tricky, reclamation systems are shown to be pretty solid by the first book, but who knows what of that was advancements after the initial colonization. There are also some ice deposits on Mars that could help.