I'm learning some common Japanese phrases and expressions from watching Anime and reading Manga.
Same actually, but more of the latter. What I do is machine-translate manga that I want to read.
I use OCR (Optical Character Recognition) software to parse Japanese characters within their bubbles and then modify the characters, if need be, through the editor. Obviously along the way I translate these using Google Translate and Translation Aggregator. Once I get to the end, I just make a PDF where I can read and select text along the way to translate as I read so I don't have to repeat this process over and over. Pretty neat.
And I know what you're thinking, ew Google Translate, right? But Google Translate isn't all that you do. There are many sentences that seem off, some phrases that seem incoherent, or even some assemblies of characters that are untranslatable. That's why you're a human on the internet: research. Online translators aren't very fond of onomatopoeia or slang, so those are easily googleable. Some authors like to add some immersion in their works in the form of dialect, and you can account for those differences through simple googling and memorization.