This is 4 novels, originally published independently( from the late 50s to early 60s), the first 2 being self-contained, although serving as prequels (potentially, originally, but explicitly here, in this edition). These are: 'They shall have stars' (also published as 'Year 2018'), 'a Life for the Stars', ' Earthman come home', and ' a Clash of Cymbals'. They were originally published out of sequence, although the fourth novel, a Clash of Cymbals, was explicitly a follow up to 'Earthman, Come Home'.
The third novel, 'Earthman come home', is the place to start and, frankly, stop if you don't like it, or even if you do, but find it enough to be going on with - I read it three times, over many years, before I was aware of the others. Don't look for any refinements of style - in fact, it's wincingly awful sometimes. Blish is definitely 50s cold war American comic book in outlook and depth of characterisation, but that has it's attractions for me - simplicity and focus on a story, and an easy read. I read all his books I could find when I was younger, for all that. I think Blish, like many of the early SF pioneers, became a regular contributor of scripts to Star Trek. Anyway, 'Earthman, Come Home' is an amusing flight of fancy. Earthmen had long ago spread throughout the home galaxy. A technology (spindizzy drives) had been developed, which enabled entire cities to lift into space ( Blish obviously had a background in engineering or physics - there's a sprinkling of plausible, if dated, technological language). Earth's cities had long ago been deserted by its cities, which traveled the galaxy looking for work amongst the settled planets, contracting for work as needed. Earth, which had once had an empire, but which few of the flying cities had now ever seen, still maintained a police force, which enforced contracts, and generally policed the flying cities. There are some nods to John Steinbeck's tale of 1930s depression America, 'the Grapes of Wrath', about migrant workers, driven by desperation to seek work in California, known as 'Okies' (from Oklahoma) - so we have 'Okie cities' and an 'Okie jungle' (hobo jungle). Don't expect Steinbeck, though, or even Philip K. Dick. The central character is the mayor of the city once known as New York (just the Manhattan bit, though). The first 'adventure' is a bit confusing, but after that it's quite straightforward.
I wouldn't be surprised if there is a film based on 'Earthman...' soon, now that the CGI is up to it.
The fourth novel continues the story, not very remarkably as I remember. The first 2 novels are set in time before the cities took off from Earth, recognisably our world a little in the future (as imagined in the 50s or 60s). 'Year 2018' centres around the bizarre effort to build a bridge from Jupiter to one of its moons, which tested a lot of the theory later used to lift the cities. 'A Life for the Stars' is the story of how the central character of 'Earthman...', John Amalfi, mayor of Manhattan-in-space, took his first step from Earth into a life in the stars.
I can understand the readers who found this a drag, though, which is why I say: read 'Earthman, come home' first - it's the most action-packed and imaginative, and is the core of the quartet; then read the sequel(if you want), 'a Clash of Cymbals'. Then read the other 2 at your leisure. They were published originally as self-contained novels (except for the last one) and should be treated as such; Blish's style is fine in short bursts, as originally conceived, but to read all these books in a row would tax most people who don't already have a fondness for them - in fact, I myself have never read them all in a row like this.
If you can get one of the old Mayflower paperback editions of 'Earthman, come home', the jacket would be worth the price: the cover is, I think, the work of cult artist Richard Powers, who was the biggest influence in jacket design for early SF, for a while. The cover has an abstract expressionist painting (paint splashes), Jackson Pollock style (I saw one of his paintings recently that looked like it might have been the one that gave Powers the idea for this). In the middle of the paint whirls and doodles is a little picture of a man in a space suit, as if in the midst of some cosmic quantum soup - for me a perfect illustration of the contents with it's 'Dirac transmitters' and 'spindizzy generators'. The uniquely American art (and some say CIA sponsored - but that's another story - see Serge Guilbaut's ' How New York stole the idea of modern art' for a start) of abstract expressionism, married to the science fiction of the cold war space race era, perfectly captures the zeitgeist of the times.