There also is the excellent book "Visual Differential Geometry and Forms" by Tristan Needham, which takes a similar visual intuitive explanation style. Also it features pictures of fruit pierced by lots of toothpicks.
I recently came across the existence of Penrose string diagram after curiously and naively researching the existence of index free notation for linear algebra. This seems to be a very interesting paper in the very same category of things I'd love to study and learn about but probably won't ever find the time!
You will possibly like Needham's "Visual Differential Geometry and Forms" and even if you don't have much time to study it, it's quite beautiful to look at from time to time.
This looks pretty good. Minor gripes: quite a few typos, and the pictures are often a couple pages away from the page that refers to them, making reading a bit inconvenient if your screen fits only one page. I only have a monochrome printer but I might spend a few bucks on a color printout at the local copy shop.
Now, that is a noteworthy weekend reading. Thank you!