http://doc.rust-lang.org/nightly/std/primitive.str.html
It's unfortunate that most of the links navigate you to the less useful doc.
You can either remove "Source Code Pro" from your box, or use Tampermonkey / Greasemonkey to remove it from the markup.
To take the given example, I would start at http://doc.rust-lang.org/std/ and hit `s` to start a search. Searching for `str::lines` gives http://doc.rust-lang.org/std/primitive.str.html#method.lines as the second result, which has the two methods conveniently next to each other, succinctly explains them, and shows an example of using each one.
I've had good luck with docs for the few third party libraries I've tried so far too. The recommended doc generation creates pages in the same format as the std docs, so as you get comfortable navigating them you get comfortable navigating many third party docs as well.
The specific issue you're talking about is due to using Markdown rather than ReST, basically. We didn't manually link those methods, but probably could.
In general, the API docs could use a lot of work. There's always so much more to do.