How far along is the new documentation? Would it be better to read https://doc.rust-lang.org/book/ or the github.io page? Maybe both?

Both would be ideal, if you have the time. The new book is far enough along that you can learn the most basic stuff from it, and the intermediate bits are coming along. The more advanced stuff is only an outline. So if you start with new, and then switch to old, you'll get the best of both.