Illustrator is your drawing program and it's good for creating logos, concepts, and digital illustrations. You don't have to have drawing skills, but it does help to take a drawing class because you will need to understand how to create images with perspective, lines, value, form among other things. There are so many basics you will learn in a drawing class that can help you with using Illustrator.
Just because you know how to use the software, doesn't always mean you'll be a good illustration.
Case in point: I was tutoring a student how to use Illustrator because he wanted to create his own comic book artwork. That was no problem. However, when I gave him projects to work on during the lesson using Illustrator, his digital drawing skills were subpar because he wasn't drawing every day. Drawing does help you become a better illustrator. Take a look at Todd McFarlane. He's been creating comics by hand since he was a kid, and eventually he went digital.
I would also NOT recommend using the trace tool because it's a waste of time and does NOT accurately give you the desired image you are looking for. You're going to have more control over your artwork using the pencil or better yet, the pen tool.