Philosophy courses did, seminar-style analytic philosophy classes in particular. (I wouldn't say that history of philosophy classes altered the way I thought, though I can totally see how Hume might be shocking to someone very new to the subject.) Aside from the actual content I learned, I got the following out of them:

1. The mental habit of condensing complicated lines of reasoning into minimal, fairly linear syllogisms, so that all of the logical dependencies and likely points of failure among the premises/inferences become much more obvious.
2. Relatedly, an eagerness to search for ambiguities in arguments and to enumerate all their possible disambiguations, with an eye for the most charitable/defensible contenders.
3. An appreciation for fine distinctions underlying seemingly straightforward concepts. (E.g., there are several related but distinct concepts that map onto the notion of a word or sentence's meaning.) These often have unexpected implications and/or vitiate seemingly plausible inferences.
4. Not being allowed, on pain of embarrassment or a bad grade, to get away with BSing or relying on unacknowledged, controversial assumptions. You have to be up-front about precisely what you mean and what's at stake.
5. Realization of the extreme rarity of knock-down arguments for any view, and the subsequent adjustment to the fact that assessing pretty much every philosophical question involves a robust trade-off of good and bad consequences. Sometimes every view on the table seems to imply something crazy, and you have to learn to accept that. And to accept that sometimes reality is crazy. (Yes, I know that the map is not the territory, etc.)
6. If arguments are soldiers, then at least learning to let some of your soldiers die - and sometimes even putting them out of their misery, yourself! It's very common in philosophical writing to go through all of the failed arguments for your view before moving on to the ones you find more promising. Even then, it's expected that you highlight their most vulnerable spots.
7. Epistemic humility. I'm much slower to draw hasty conclusions with high certainty on a given topic before I find out the best of what all sides have to offer. I definitely still form fast and intuitive judgments before investigating disputed subjects deeply, but I don't pretend that they're likely to be the last word or even novel contributions lacking high-level criticism.
8. A much richer sense of the space of philosophical views. But then again, probably something analogous holds for most other disciplines (biologists are presumably better-tuned to the space of biological hypotheses). Still, though, philosophical-view-space intersects an unusually large number of things.

I don't know if you're in need of any of these things, or if you're likely to acquire them through a small handful of philosophy classes. Even if you are, whether or not you'd succeed greatly depends on the quality of your teachers and classmates.