The sheer volume that is capable of this program astounds even me and I've been using this program since 2009. The ability to create, to adjust, to layout, and to even edit. I've used Illustrator for Logos, Adverts, Websites, Motion Graphics, Newsletters, Brochures, Stickers, Buttons, T-Shirts, Posters, Signs (larger and small) and even once- A Hoodie for a Dog. I always recommend Iusing other programs like InDesign and Photoshop in tandem, but my main program is Illustrator.
The Live Trace feature needs some work, big time. It needs to be adjusted so when you make more subtle changes, they actually work. It's always a roll of the dice if it works the way that you're hoping for.
Start slow, find a specific project you want to do and read up on a how-to. YouTube and the internet can be a big help out in a pinch if you get stuck. Just look it up and you'll be an expert in no time.
Logos. In almost in case you can think of; Recreating, Live Tracing, Making from Scratch and Adjusting.