So how did the redhaired chick learn their language?

I followed the language part, but they did just allude to her "just figuring it out" all that vocabulary, from unknown markings, there is no way. When we learn languages of other nations, one learns the basics of ours just as they tried to show in the film, the difference is, other humans don't use alien guttural noises and ink blot writing, they have usually have the same vocal anatomy and pronounce the worlds we tell them, and vice versa, and even that is a slow process, I'm talking no contact at all. There was no way we could decipher their written language without vocal communication, vocal is typically first, with few exceptions like the kemetian writing, which has been falsely translated by westerners for so long.

Yeah, that's one complaint I have. The movie didn't make it clear how much time had passed so I was quite surprised how quickly they were learning the language, especially when Louise entered the ship by herself and could just read their language by eye without the aid of a computer. Was it supposed to imply that she was understanding it via telepathy or something?

You're right. The movie clearly needed a montage set to 80's pop/rock music showing her trying to learn, finding it difficult, gradually understanding, and finally Understand their language.