Charles Staats III wrote an very handy tutorial on Asymptote from the point of view of an Asymptote beginner who may already know some LaTeX and TikZ, though that’s not necessary.
It starts with choosing the output format, showing simple commands for printing text, and explaining how key=value arguments work in Asymptote. It continues with easy two-dimensional drawings, explaining how to adjust image size and draw lines, add arrowheads, draw curved paths, and add markers. It continues generating circles, ellipses, arcs, and polygons and explains the concepts of filling, clipping, and transformations like shifting, scaling, and rotating, plus more advanced topics.
In the next chapter about three-dimensional images, Charles explains what interactive images are and how to render good-quality rasterized images. He continues by using three-dimensional paths, projections, and lightning. He shows how to generate various kinds of surface plots and how to use labels and arrowheads in three dimensions.
The tutorial includes an appendix on installing and updating Asymptote for macOS, Linux, and Windows.
It’s a very recommendable tutorial, much easier to read than the original manual. You can find it here: asymptote.sourceforge.io/asymptote_tutorial.pdf