If you have an IQ in three digits and an analytical mind, and want to learn Italian as fast as possible at minimal expense, then this book would be my first recommendation.
It was first published in 1943, and thus uses the grammar-translation method of teaching. This is how I learnt French and German in school, and it works for me. It did not, though, work for about 60% of my classmates, who at the end of four years were able to recite tables of German verbs, but not to put a sentence together. Which is no doubt the reason for the modern revision in methodology.
If you learn best by systematic explanation and analysis/synthesis (as I do), then, this (the Speight version) may suit you. If you learn better by imitation and immersion, you may do best with the the modern edition by Lydia Vellacio and Maurice Easton. The latter, however, has corresponding CDs, which the 1943 edition (unsurprisingly) does not; so if you have no live source to talk Italian with, this may be the major consideration.
Indeed, no book, however thorough, can teach you to understand foreigners when they babble at you, and so none can be adequate on it own. But I would still advocate this book as a first step, to get the grammar and a basic vocabulary in your head, before proceeding further.
If money is no object, there is Prego!, but it is a textbook for university courses, and makes numerous references to “your instructor”. It has a plethora of materials with colour illustrations and audio CDs and so on (at a total cost of $many); but I find it rather sloppy, particularly in distinguishing aspects of pronunciation such as open and closed e (as in English pet/pate) and o (cot/coat). Ms Speight is much more careful.
Should you decide to go with the Speight version, there are a couple of fairly obvious caveats:
• Lire are now euros. • Italians are pretty slapdash about diacritics, much more so than the French or the Germans, and there are dialectical differences. Ms. Speight uses the acute accent over i and u (e.g. lunedí, piú), whereas most dictionaries seem to use the grave (lunedì, più). She also mentions the use of the circumflex to indicate an omitted letter (e.g. studî, from studii, plural of studio): this is now considered fussy and/or old-fashioned, much as writing rôle rather than role would be in English. These are, however, trivial matters.