There are no wonders. If you want to move large cities underground, you have to dig. A lot.
Radon concentration and similar things do exist, but - except some very few areas on the world - they are negligible. Although the highest purely natural radioactivity of the world is far higher as any prescribed health standard (around 0.8 Gy/year, living there means a much higher risk as smoking).
People doesn't really like to move underground, but if they have enough motivation, they will do. You don't need some cataclysmic event for that, it is enough if they can find work only there.
A radioactive surface isn't enough motivation, because in this case, also the ground water will contain a lot of solved radioactive materials, mainly metal salts, which doesn't make the situation better underground (while the increased costs of the digging still exist).
The main problem with digging, that it requires a lot of people, working a lot of dirty work, with big and costly machines. Consider the costs of the underground train tunnels. The Eurotunnel (binding UK and France) costed around \$21billion. The cost of similar underground tunnels costs mainly between \$10million-\$30million for every km.
Building the big and costly machines requires also a big and costly machine industry.
The better usability of the free, now empty surface land isn't a big advantage, because most of it is already uninhabited, even in largely overpopulated countries. We, humans, tend to concentrate ourself into big cities.
About the actual costs and technologies of the digging there is a quite useful SE site.
In any case, the opencast building (i.e. digging a big hole from the surface) is always much cheaper as digging essentially large mines.
Although heating won't be a high cost of these cave-cities, their cooling may be much bigger. And also their air ventillation. There are no wonders also in this case: you have to circulate water and air between the surface and (if it is irradiated, a closed water loop and air filters are adviced). You have to use pumps and ventilators for the task.
Note: there are radioactive gases, some of them won't be filtered by anything (noble gases). Although most of them decays very fast (thus they long decay as the caves will be built), or very slow (thus they aren't very radioactive). Fortunately, practically none of them is produced by normal fission processes (nuclear energy production + atomic bomb explosions), thus we have luck.
Fire, and any carbon dioxide producing objects (non-electronic cars) would mean much bigger risk and cost (air).
Energy production will still have a lot of contact with the surface, because there is no energy production without heat production, and you have to do something with the heat. Of course you can use surface solar panels and windmills to produce the energy on the land. In this case you will have a lot of cables.
The whole society will be much more strongly dependent from the technology, which will elevate its worth. The probable result will be that the law and the customs will follow the same. Smoking will be much more evil as now, destroying any technology, particularly if it has anything to do with the life-sustaining systems, will be much more hardily punished.
Working on technology (engineers, scientists, teachers of these) will be probably much more honored as today.
Sustaining the civilization will require much more work. It will have a negative effect to the life quality.
The civilization would require around 30-50 years to adapt (and, to reach the life quality of their old surface again).
If the move to underground is the result of a hard pressure, then the first years will be crucial and for the large masses, mainly catastrophic.
After such events - revolutions, downfall of a world system, world wars, etc - the society needs around 1-2 decades to stabilize.
I can advice two very interesting novels from societies in similar situations: