I understand that this won't be a magical transformation, and I don't expect fluency in the short time that I have to study. I am really just looking to be able to communicate on a somewhat basic level.
Ok, let me suggest this. Start with Pimsleur levels I and II if you can find them at a local library. They are too expensive to be worth buying, IMHO. These use the spaced repetition method (eg. ask you to respond with something you just heard, then progressively stretch out the time you are asked to recall it, just like Anki or any other SRS does). They say do each lesson once, I found I got more when I did the lesson once in the morning and again in the evening. Progress is kinda slow for the time invested, so I wouldn't go past the 1st 2 sets. But what you have learned, you will know and effortlessly recall instead of having to flip thru a phrasebook.
Next, get the Michel Thomas Italian. They keep repackaging the series (see https://en.wikipedia.org/wiki/Michel_Thomas_Method#French.2C_German.2C_Italian.2C_and_Spanish ) but what you want is the set covering the 1st 8 hours or so. Similar in some ways to Pimsleur, it takes you a lot farther in terms of being able to express thoughts and ask for things although it is skimpy on vocabulary.
After this you can go on and use the advice from Wyner's book, but I think the approach above gives you a reasonable base. After doing this we were able to travel on the Adriatic side of Italy, where English is not common, and get by (eg. rent hotel rooms, basic transactions in stores and restaurants, etc).
My experience has been that at this level communication is more one-way. You can ask for things and people will understand, but responses spoken at normal speed are hard to understand and any real conversation is impossible because you don't have the vocabulary of even a toddler, nor can you parse the words to even find the ones you know. This may sound discouraging, so I want to make clear I'm not saying its not worth trying, not at all, but getting by is different from being a temporary part of the community and chatting with local folks,with the merchants at the farmers markets, etc.