I've had to learn quite a few in my life, but the one I learned the fastest was Italian. That is simply because I used the best method.
The method was to move to Italy (Rome specifically) and live in a house filled with 200 people none of whom could speak my native tongue. However, that alone would not be enough.
The first trick was watching cartoons in Italian. Rome had about 30 TV stations at the time. I completely ignored the three RAI channels (the official national channels), but found one that was showing all kinds of American shows in Italian. Since I had arrived from Slovakia, all those shows were completely new to me, so I would recommend a different channel for someone familiar with American shows.
Anyway, the station also showed a lot of Japanese made cartoons. They were not obviously Japanese - all characters were whites and blacks, the only unusual thing about them was that they ate their lunches with chop sticks.
One cartoon series in particular was about a ragazzo diventato robot (boy who became a robot). The basic idea was that the boy's father designed some robots that were activated prematurely during a storm and turned against humanity. The only way to beat them was for the father to turn his own son into a robot so he could fight them.
The cartoons were aimed at children, so they used very short sentences. I quickly learned my very first two Italian phrases: "Uccidetelo!" and "Eliminatelo!" They did not turn out too useful after my enrollment at Gregorian University studying Canon Law, as they translate: "Kill 'im!" and "Eliminate 'im!" respectively, while the Church no longer does that. But it was a good start.
One might thing that Perry Mason would have taught me more useful phrases, such as "Signori e signore della giuria", i.e., "Ladies and gentlemen of the jury" (or, actually, "gentlemen and ladies"), but it turned out Church courts do not use juries.
But watching La Famiglia Bradford (a.k.a. Eight is enough) helped me master everyday Italian.
Later I was able to actually watch full length original Italian movies and learn some very useful phrases. For example, I still remember from Giordano Bruno that a good diplomatic letter requesting a favor from the Pope should start with "Mentre baccio la Vostra pantofola sacra" ("While I kiss Your sacred slipper").
But I really graduated into accelerated learning of Italian when I always ended up sharing the dining room table with a real jerk who prided himself about being a bilaureato (a person with two doctor's degrees), and thought no one was smarter than him. We ended up in serious theological arguments twice a day: At lunch and supper.
He had one big advantage: He was an Italian and we argued in Italian. You'd be surprised how fast I learned Italian just to make him lose!