I'm new to the book, and not sure where to start. How should I as a writer use this tool to improve my writing?
The purplest prose comes from writers who rely on a thesaurus to choose synonyms, without fully understanding nuance and context. It's a bit like equipping your car with every available gadget from the local auto parts store -- the effect is rarely harmonious, and often unintentionally hilarious.
Better to read Roget in the bathroom than use it as a reference while you're writing. You'll expand your vocabulary in a nice, general way, safe from the temptation to over-accessorize your writing.
The wrong synonym creates an impersonal level of abstraction that distances you from, and subtly annoys your audience. Better to write honestly, using the words you already know, while striving still to increase your knowledge and appreciation of the language.
Careful diction creates believable characters: the naive but earnest social climber who tries to impress with a not-quite-right word choice; the persuader who uses words to orchestrate emotional, rather than intellectual agreement; the warm, likable character whose words connote wisdom gained more from experience than from formal education.
Really good writing, whether fiction or nonfiction, comes from those who are comfortably intimate with the language. The right word choice depends not only on definitions, but also fluency, emotion, rhythm, harmony, diction, and character.
A thesaurus is helpful when the word you want is "on the tip of your tongue." I look up whatever word I can think of that is closest in meaning to the word I want, and then from there, I "surf" the thesaurus, flipping from word to word, coming closer in approximation to the desired word. I find the exact word I had in mind probably 80% of the time.
A thesaurus isn't really a book of synonym lists, and you treat it as one at your peril. It lists words that might convey the shade of meaning you're looking to express. But they're just as likely not to -- more so on average. It's up to you, the writer, to hear through the static.
As Kate noted, one way to never use a thesaurus is for learning new words. The way to learn new words is to read, read, read.
I don't use it. I would caution against others using it, or any other thesaurus.
I think that a large, varied vocabulary is a great thing, but the problem with gaining your vocabulary from a reference source rather than from reading prose is that you don't really get the more subtle meanings and shadings. Somebody once said that there's no such thing as an exact synonym, and while I'm not sure if that's true, I definitely think that thesauri go way too far in claiming that words are synonyms when they really aren't. I just looked at the word 'home' in an online thesaurus, and there's not one of the suggested synonyms that would work even in a simple sentence like "I want to go home," and some of the suggestions are just plain wacky. (http://thesaurus.com/browse/home).
I think that thesauri treat writing as if it's a science, when really it's an art, or at least a craft.
ETA: But after some thought - maybe a thesaurus would be useful when you just need your memory jogged. Like, not as a way to learn whole new words, but rather as a tool for times when you know that the word you've used isn't quite right (either because of repetition or because the connotation isn't quite what you're looking for), and need suggested replacements. I can see a thesaurus being useful in that context, as long as you restrict yourself to only using words with which you're already familiar.
So, with that understanding - I'd say the best time to use it would be when you're reading over your work, and find a word that doesn't quite work. I wouldn't use it for first drafts, when you're just trying to get the big ideas down, but I can see it being useful for fine tuning. Interesting.