Take a look at the links on my About page. There's a link to an Illustrator Pen tool exercise and another to a PDF with some notes on taking data out of ArcGIS and into Illustrator.
Other than that, I'd suggest just getting familiar with Illustrator in general and not focusing on special cartographic use, at least initially. You need to become proficient at using the pen tool, the selection tools, playing with colors in the color palette, setting up layers (similar to ArGIS). You'll eventually want to learn to make your own swatches, and a symbol library. Knowing how to place and scale images and use some of the special effects functions like drop shadows, feathering, or transparency.
After the basic pen tool exercise from my site, I'd suggest placing a simple map scan or image in an empty AI doc and recreating it by tracing line work into new layers and setting line styles.