The Geoscape dome is the wonder thats unlocked as the last research in the Biotech tree. As long as you don't spread your research to evenly over all of the trees you should easily be able to unlock this by around Sol 75.
Make sure to scan all anomalies and do the space anomalies that increase research.
The USA recieve quite a bit of sponser money so you can always outsource some research to speed this up as well.
The Megamall is unlocked from the beginning and you'll want to be putting at least 1 into every dome (I was putting 3+ into mega domes) as they fulfill all comfort needs for colonists.
I suggest playing with the modifiers prefab colony, fast scanning, fast rockets, increased production & extra applicants. No disasters or events is optional depending on your preference (Both can help or hinder)
None of these options disable achievements.
I did this in the same playthrough as
And managed to unlock everything in both research trees and build the Geoscape Dome by Sol 95, so its tight but doable to get both in 1 playthrough.