My copy of this is the first omnibus from 1970 -- my first book from SFBC -- and I really loved it back then when I was 15. Now? Too lecture-laden and very boring. All of us who read it back then don't like it now and I was the only one in the 12-person group who actually finished it.
There's four short novels and they are in chronological order in the omnibus, but they weren't written in that order. The first book isn't bad, and they discover anti-gravity and the machines that do it are called "spindizzies"; they also discover an anti-agathic med so people can live as long as they have the meds and don't get killed. The first colonists go out. At this time, the Believers are gathering to be saved.
Earth loses the spindizzy and then finds it again and the next three books are about what happens to New York, New York when they take off. They planned to do work for people on planets, but that doesn't turn out well, and then they have to fight back a fundamentalist religious group -- the Web of Hercules -- before the high-level city people die and make new universes. They think.
It's rare for us to agree at this level. The people who didn't finish the book couldn't stand reading more (books were thrown) and I finished it but didn't like it any better.
(Just imagine the jokes and laughs we had with Believers and the Web of Hercules on May 21st!)