The following protocol is very dumb, and relies on a lot of mental brute force, but I find that it works very well indeed.
First, learn the alphabet. the most basic survival phrases, the phonetics, the basic grammar. This may be the steepest part of the learning curve. I recommend that you get at least an introductory book to help you through this phase, the A1 level.
Now comes the fun part.
The key element to accelerated and efficient language learning is wanting, needing, craving to understand and to be understood. Choose a text, any text, which you know will trigger all your "I want to know what it says!" instincts. I myself find that highly dramatic works, with lots of suspense and high emotional torque, are ideal. Take a dictionary and just look up every single word that you don't know. You write each one of them down, with, first, the exact pronunciation, then, their definition in the original language, the translation to your own language, and, optionally (and I do recommend taking that option), a couple of examples of its use, better if you come up with them yourself.
It's very intense, and one can get deeply immersed in the flow, so pay attention to the clock: there's a very high risk of Tetris Effect/Just... One... More... Word... effect taking place. Like going to the gym, you need to pace yourself: if you go everyday for two weeks and then give up for three months, we won't be achieving much. Keep it at six hours weekly maximum if you're doing this on the side, three hours is a reasonable rate.
The next step would be, once you're fairly confident you won't make a fool of yourself, to join a forum where topics you care a lot about are discussed very seriously, and then trying to contribute to the discussion. This will force you to write a lot, very quickly, and your interlocutors will be very unforgiving of mistakes, so you'll be very motivated to check and double check. Giving a teacher a sloppy piece is simply laziness, giving it to a discussion board is an affront.
I focus this much on the written media because that's where you'll get the most information bandwidth, so to speak, and because it's both easier to pick apart and to put together than the oral language, kind of like the difference between a turn-based game and a real-time strategy game. Subtitled movies can be a fun, low intensity tool, but the difficulty when learning languages other than English is to find versions with good subtitles (you DO NOT WANT the bad subtitles).
But you may well have to learn to speak the language properly, and one of the fast ways of improving one's ability to learn languages is, well, singing. Usually there's opportunities to join a choir for free (especially if you're okay with church-y environments), and they're a wonderful learning opportunity for beginners. The more you master voice and rhythm, tone and timbre, the easier you will find it to attune yourself to any particular language's set of sounds and paces.
And know this: the more languages you learn, the easier it becomes to learn more. Additionally, the same way learning programming languages helps you clarify the way you think and pattern your thoughts more logically, learning new human languages will give you a firmer grasp of the metaphors on which it is all built, and a better understanding of both universal human psychology, and your own language (like Monsieur Jourdain, who found out to his great amazement that, his whole life, he had been speaking in prose... but, hopefully, what you find out will be more meaningful in all sorts of ways).
Of course (I think I should have pointed that out in my first post), but physics/math/etc also take a long time to learn properly, so the time required becomes much less relevant.
I do not mean that one should necessarily learn a new language instead of learning math - although I might say that if you already know a lot of math (enough to get significant benefit to your thought processes), it might be useful to spend some time learning something that trains different aspects of mental processing (like learning a language). If I had to speculate on what any specific benefits might be, I would suggest that it comes from having more than one independent lens through which you interpret the world/more than one basis for your thought processes, and the benefit to mental flexibility you can get from switching between them (I'm not sure if I am properly communicating what I mean by that though).