First of all: if you are someone who hasn't read a lot of books about the English language, but you have to do a fair amount of writing in your daily life and you feel that you need help with grammar and punctuation, this book is probably not for you.
Gowers' original book was written in the late 1940s and early 1950s, mainly for use by civil servants who were already highly educated but who needed reminding that they were supposed to be servants of the public, not masters. Gowers' advice is mostly about cleaning up bad and pretentious English, not about basic points of style and usage for people who just don't know how to put a sentence together. If you want a useful book which will tell you things like where to put the full stop in a sentence that ends with a quote, then you need a good style guide. The best one for my money is the Oxford University Press 'New Hart's Rules', an excellent handbook of guidelines on basic usage, clarity and good plain style. Alternatively, you could get Fowler's 'Modern English Usage' but the current edition of it (edited by Robert Burchfield) is somewhat controversial, and previous editions, though fascinating, are a bit out of date. 'New Hart's Rules' covers the same ground but is shorter and more handy for everyday use.
'The Complete Plain Words' is a different sort of book from any of the above, and is really intended for people who already think they know how to write. It's extremely good at clearing up common confusions in the minds of educated people, such as the distinctions between 'abrogate' and 'arrogate', or 'comprise' and 'compose' (or, for that matter, the specific difference between 'comprise' and 'include'.) If this is not the kind of thing you are looking for, then you probably need 'New Hart's Rules' or Fowler. Everybody else needs them too, but this book is actually a fairly specialised guide to writing official English, and is aimed at people who do that for a living.
For those people, Gowers is a great and enlightening read. My own copy of the book is the second edition, edited by his great successor Sir Bruce Fraser, and apart from its value as a guide to good usage it's an interesting snapshot of the state of English in the early 1970s, when Fraser carried out his revision. It's also wise, witty and full of fascinating examples of what the author and editor considered good as well as bad English.
There is a current fad for correctness in language, of which the bestselling example is probably Lynne Truss' book 'Eats, Shoots and Leaves'. It has been hyped as a useful and accessible guide to correct English (or at any rate, correct punctuation). I haven't read it, though, for two reasons: one is that a quick flick through the book revealed that it didn't contain anything that wasn't already in any of the books mentioned above, which I already own; and the other is that if I were looking for someone to teach me about how to write clear and unpretentious English, I certainly wouldn't go to a journalist.
To sum up: if you feel lost writing English and want to know how to avoid basic mistakes, get 'New Hart's Rules' or Fowler's 'Modern English Usage'. If you feel like you know how to write but find it difficult to make your meaning clear, get them anyway, but also get this.