Learning any language changes your thinking patterns to the implicit assumptions built into the language structure. you may find the work of Ferdinand Saussure of interest.
Just a simple example of learning Dutch, which blew my mind.
1. Monday in Dutch is maandag, which means 'moonday'
2. Friday in Dutch is vrijdag, which means 'free day' (and is pronouced 'freydag')
Obviously the French is lundi, but I never made the connection with lunar.
So for 28 years (4 year's worth of Moondays) of my life I never considered the connection between Sunday, Monday and Moonday.
Perhaps it's not scientific fact but actually learning another language fluently and seeing the differences was pretty freaking awesome to me.
There's other fascinating differences like the difference between 'Icing sugar' in English and 'Flour suger' in Dutch, one explains what you do with it vs the other explains what it resembles.
Perhaps it only challenges simple brains like mine.
Learning new languages can definitely make you better-informed about language (for example about etymologies, like you mentioned, and about different ways of doing things with language on every level from phonology to pragmatics). When I studied Portuguese after Latin and Arabic after Hebrew, I was making connections between languages all day long, due to their historical relationships. Many of those connections would be missed by monolingual native speakers, even though they have much more overall ability in the languages!
It's also interesting to see how different languages give you different options. For example, Latin has infinitives that can inflect for tense and voice, so you can say "to have written" or "to be written" in single words, not just "to write". Portuguese lacks these, but has infinitives that can inflect for person and number, so you can say "for us to write" in a single word. English has neither! This is just scratching the surface of grammatical variation within closely related languages.
I think the controversy alluded to in this thread is that there is an old, fascinating theory that people think qualitatively differently when using different languages, or else that some languages are more effective at or conducive to thinking certain thoughts. (One form of this idea is called the Sapir-Whorf hypothesis, even though it's not clear that Sapir and Whorf understood it in exactly the same way.) There are many suggestive examples in support of this idea, but the strongest forms of the claim are in disrepute in the academic linguistics community.
The Sapir-Whorf sorts of issues are potentially rather different from saying that you'll understand language better by learning more languages, since they suggest even that you may understand the world as a whole differently.
I studied German and Japanese and it was very interesting indeed. However, what I was responding to is the common misnomer that language studies create cognitive improvements as my high school advisor and the above poster alluded to. The scientific consensus at the moment is that so far it only appears to help in word search ability. There's no other established improvement.
