If you ever intend to move to another country earnestly (i.e. to actually live and work there for a significant time, not only for a few months or a year), I certainly do recommend to learn the language as well as possible. Definitely well beyond the level required in the coffee kitchen.
I work in a German IT company (but a subsidiary of a global concern), and we have a great many employees from abroad; both as employees locally, as well as frequent partners in mixed-shore projects. Yes, we use English wherever necessary, but at some point there always are situations where failing to have a rock solid grasp of the local language is difficult for all involved. In IT, with our skills shortage, it's not really an issue, but in an industrial sector with less splendid job chances, this might just be the tipping point where your favourite company takes the other applicant who may not be as great as you, technically, but who has a firm grasp of the language.
Regarding when to learn that language, I'd say as early as possible - so as soon as you know that there's a certain chance that it may happen. The effort you would put in should of course scale with the certainty of you ever needing to actually use it. But starting early means you'll be younger (everything is learned easier when you're young), have more time until you need it, and can learn with less pressure (which is very valuable). You might simply start viewing movies and reading books in the language; things like that.
If you just want to open up possibilities, you can also simply pick any language of a country you like for vacations - even just chatting with the locals can make a stay that much more enjoyable.
Finally - after 3 or 4 languages, learning even more seems to get easier and easier, as you can transfer knowledge from one to the other to some degree. So even if you don't actually need that additional language now or ever, if you ever should come into a dire situation where you must learn a further one, you might be very thankful.
All of this assumes that you find something, even if it's just a tiny aspect, that is somewhat interesting to you. Do not waste effort on learning something you're not interested in whatsoever - that would just be futile anyways.