Adobe Illustrator is an immensely powerful vector graphics drawing program. It can be intimidating for first time users.
I recommend any book in the “Missing Manual” series http://missingmanuals.com/, and the Peachpit Press “Non-Designer’s Book of” series by Robin Williams http://www.peachpit.com/search/index.aspx for people new to graphic design.
Adobe has structured training in their “Classroom in a Book” Series http://www.adobe.com/training/books/
There are lots of Adobe Illustrator tutorials and videos available on the Web:
Lynda.com (high quality paid tutorials) http://www.lynda.com/
nDesign Studio http://www.ndesign-studio.com/resources/tutorials/
Smashing Magazine http://www.smashingmagazine.com/2009/01/04/50-excellent-adobe-illustrator-video-tutorials/
Free Illustrator Tutorials http://www.adobeillustratortutorials.com/free/index.php?cat=1
For more advanced work, http://vector.tutsplus.com/
Keep in mind that there are many resources, and there is no “best” choice because it depends on your learning style which ones will work for you. Also check your local community college, school districts and neighborhood centres, they often have beginners evening classes, if you learn best in a classroom environment.